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Topic 13 Area TOPIC OVERVIEW VOCABULARY 13-1Areas of Parallelograms and Triangles 13-2Areas of Trapezoids, Rhombuses, and Kites 13-3Areas of Regular Polygons 13-4Perimeters and Areas of Similar Figures 13-5Trigonometry and Area DIGITAL APPS English/Spanish Vocabulary Audio Online: EnglishSpanish altitude of a parallelogram, p. 520 altura de un paralelogramo apothem, p. 532apotema base of a parallelogram, p. 520 base de un paralelogramo base of a triangle, p. 520 base du un triángulo center of a regular polygon, p. 532 centro de un polígono regular composite figure, p. 520 figura compuesta height of a parallelogram, p. 520 altura de un paralelogramo height of a trapezoid, p. 526 altura de un trapecio height of a triangle, p. 520 altura de un triángulo radius of a regular polygon, p. 532 radio de un polígono regular PRINT and eBook Access Your Homework . . . Online homework You can do all of your homework online with built-in examples and “Show Me How” support! When you log in to your account, you’ll see the homework your teacher has assigned you. Your Digital Resources PearsonTEXAS.com Homework Tutor app Do your homework anywhere! You can access the Practice and Application Exercises, as well as Virtual Nerd tutorials, with this Homework Tutor app, available on any mobile device. STUDENT TEXT AND Homework Helper Access the Practice and Application Exercises that you are assigned for homework in the Student Text and Homework Helper, which is also available as an electronic book. 516 Topic 13 Area 3--Act Math The Great Enlargement Today, many people take photos with a digital camera or a smartphone. Digital photographs are easy to edit, store, and share with others. Now, printing digital photos is becoming easier. More and more new printers have e-printing capabilities, so people can print directly from their cameras or smartphones. This 3-Act Math task will make you think twice about printing photos! Scan page to see a video for this 3-Act Math Task. If You Need Help . . . Vocabulary Online You’ll find definitions of math terms in both English and Spanish. All of the terms have audio support. Learning Animations You can also access all of the stepped-out learning animations that you studied in class. Interactive Math tools These interactive math tools give you opportunities to explore in greater depth key concepts to help build understanding. Interactive exploration You’ll have access to a robust assortment of interactive explorations, including interactive concept explorations, dynamic activitites, and topiclevel exploration activities. Student Companion Refer to your notes and solutions in your Student Companion. Remember that your Student Companion is also available as an ACTIVebook accessible on any digital device. Virtual Nerd Not sure how to do some of the practice exercises? Check out the Virtual Nerd videos for stepped-out, multi-level instructional support. PearsonTEXAS.com 517 Activity Lab Use With Lessons 13-1 and 13-2 Transforming to Find Area teks (3)(A), (1)(D) You can use transformations to find formulas for the areas of polygons. In these activities, you will cut polygons into pieces and use the pieces to form different polygons. 1 Step 1 Count and record the number of units in the base and the height of the parallelogram at the right. Step 2 Copy the parallelogram onto grid paper. Step 3 Cut out the parallelogram. Then cut it into two pieces as shown. Step 4 Translate the triangle to the right through a distance equal to the base of the parallelogram. The translation results in a rectangle. Since their pieces are congruent, the parallelogram and rectangle have the same area. 1.How many units are in the base of the rectangle? The height of the rectangle? hsm11gmse_1001a_t09484 2.How do the base and height of the rectangle compare to the base and height of the parallelogram? 3.Write the formula for the area of the rectangle. Explain how you can use this formula to find the area of a parallelogram. 2 hsm11gmse_1001a_t09486 Step 1 Count and record the number of units in the base and the height of the triangle at the right. Step 2 Copy the triangle onto grid paper. Mark the midpoints A and B and draw midsegment AB. A B Step 3 Cut out the triangle. Then cut it along AB. Step 4 Rotate the small triangle 180° about the point B. The bottom part of the triangle and the image of the top part form a parallelogram. 4.How many units are in the base of the parallelogram? The height of the parallelogram? B A hsm11gmse_1001a_t09490 continued on next page ▶ 518 Activity Lab Transforming to Find Area hsm11gmse_1001a_t09491 Activity Lab continued 5.How do the base and height of the parallelogram compare to the base and height of the original triangle? Write an expression for the height of the parallelogram in terms of the height h of the triangle. 6.Write your formula for the area of a parallelogram from Activity 1. Substitute the expression you wrote for the height of the parallelogram into this formula. You now have a formula for the area of a triangle. 3 Step 1 Count and record the bases and height of the trapezoid at the right. Step 2 Copy the trapezoid. Mark the midpoints M and N, and draw midsegment MN . Step 3 Cut out the trapezoid. Then cut it along MN . M N Step 4 Transform the trapezoid into a parallelogram. 7. What transformation did you apply to form a parallelogram? 8. What is an expression for the base of the parallelogram in terms of the two bases, b1 and b2 , of the trapezoid? hsm11gmse_1001a_t09492 9. If h represents the height of the trapezoid, what is an expression in terms of h for the height of the parallelogram? 10. Substitute your expressions from Questions 8 and 9 into your area formula for a parallelogram. What is the formula for the area of a trapezoid? Exercises 11. In Activity 2, can a different rotation of the small triangle form a parallelogram? If so, does using that rotation change your results? Explain. 12. Make another copy of the Activity 2 triangle. Find a rotation of the entire triangle so that the preimage and image together form a parallelogram. How can you use the parallelogram and your formula for the area of a parallelogram to find the formula for the area of a triangle? 13. a. In the trapezoid at the right, a cut is shown from the midpoint of one leg to a vertex. What transformation can you apply to the top piece to form a triangle from the trapezoid? N b. Use your formula for the area of a triangle to find a formula for the area of a trapezoid. 14. Count and record the lengths of the diagonals, d1 and d2 , of the kite at the right. Copy and cut out the kite. Reflect half of the kite across the line of symmetry d1 by folding the kite along d1 . Use your formula for the area of a triangle to find a formula for the area of a kite. d1 d2 hsm11gmse_1001a_t09495 PearsonTEXAS.com hsm11gmse_1001a_t09496 519 13-1 Areas of Parallelograms and Triangles TEKS FOCUS VOCABULARY TEKS (11)(B) Determine the area of composite two-dimensional figures comprised of a combination of triangles, parallelograms, trapezoids, kites, regular polygons, or sectors of circles to solve problems using appropriate units of measure. •Altitude of a parallelogram – An altitude of a parallelogram is any TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. •Composite figure – A composite figure is a combination of two or Additional TEKS (1)(F), (11)(A) segment perpendicular to the line containing the base, drawn from the side opposite the base. •Base of a parallelogram – A base of a parallelogram is any one of the parallelogram’s sides. •Base of a triangle – A base of a triangle is any one of the triangle’s sides. more figures. •Height of a parallelogram – The height of a parallelogram is the length of an altitude of the parallelogram. •Height of a triangle – The height of a triangle is the length of the altitude to the line containing that base. •Number sense – the understanding of what numbers mean and how they are related ESSENTIAL UNDERSTANDING You can find the area of a parallelogram or a triangle when you know the lengths of its base and its height. Key Concept Parts of a Parallelogram Term Description A base of a parallelogram can be any one of its sides. The corresponding altitude is a segment perpendicular to the line containing that base, drawn from the side opposite the base. The height is the length of an altitude. Diagram Altitude Base Key Concept Area of a Rectangle hsm11gmse_1001_t09124 The area of a rectangle is the product of its base and height. A = bh h b 520 Lesson 13-1 Areas of Parallelograms and Triangles hsm11gmse_1001_t09119 Key Concept Area of a Parallelogram The area of a parallelogram is the product of a base and the corresponding height. h A = bh b Key Concept Area of a Triangle hsm11gmse_1001_t09121 The area of a triangle is half the product of a base and the corresponding height. h A = 12bh b Postulate 13-1 Area Addition Postulate hsm11gmse_1001_t09140 The area of a region is equal to the sum of the areas of its nonoverlapping parts. Problem 1 Finding the Area of a Parallelogram What is the area of each parallelogram? Why aren’t the sides of the parallelogram considered altitudes? Altitudes must be perpendicular to the bases. Unless the parallelogram is also a rectangle, the sides are not perpendicular to the bases. A B 4.6 cm 3.5 cm 4.5 in. 4 in. 2 cm 5 in. You are given each height. Choose the corresponding side to use as the base. A = bh A = bh hsm11gmse_1001_t09134 hsm11gmse_1001_t09132 = 5(4) Substitute for b and h. = 20 The area is = 2(3.5) =7 20 in.2. The area is 7 cm2. PearsonTEXAS.com 521 Problem 2 TEKS Process Standard (1)(F) Finding a Missing Dimension For ▱ABCD, what is DE to the nearest tenth? F 9 in. D C 13 in. A What does CF represent? CF is an altitude of the parallelogram when AD and BC are used as bases. E 9.4 in. B First, find the area of ▱ABCD. Then use the area formula a second time to find DE. A = bh = 13(9)hsm11gmse_1001_t09136 = 117 Use base AD and height CF. The area of ▱ABCD is 117 in.2. A = bh 117 = 9.4(DE) Use base AB and height DE. 117 DE = 9.4 ≈ 12.4 DE is about 12.4 in. Problem 3 Finding the Area of a Triangle Why do you need to convert the base and the height into inches? You must convert them both because you can only multiply measurements with like units. Sailing You want to make a triangular sail like the one at the right. How many square feet of material do you need? Step 1 Convert the dimensions of the sail to inches. + 2 in. = 146 in. (12 ft # 121 in. ft ) + 4 in. = 160 in. (13 ft # 121 in. ft ) Use a conversion factor. Step 2 Find the area of the triangle. A = 12bh = 12 (160)(146) Substitute 160 for b and 146 for h. = 11,680 Simplify. Step 3Convert 11,680 in.2 to square feet. 11,680 in.2 ft # 1 ft 1 2 # 121 in. 12 in. = 819 ft You need 8119 ft2 of material. 522 12 ft 2 in. Lesson 13-1 Areas of Parallelograms and Triangles 13 ft 4 in. Problem 4 TEKS Process Standard (1)(C) Finding the Area of a Composite Figure Select a technique that will help you find the area of the composite figure below. Then find the area of the figure. 3 cm Could you divide the composite figure differently and still use the Area Addition Postulate? Yes, but you might not be able to use mental math to do the calculations. 6 cm 3 cm 4 cm You can use mental math for this problem, since the calculations are easy enough to do in your head. The area of each triangle is 12(3)(4) = 6. The area of the parallelogram is (6)(4) = 24. To find the area of the entire figure, add the areas of the two triangles and the parallelogram. 6 + 6 + 24 = 36 NLINE HO ME RK O The area of the composite figure is 36 cm2. WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. Find the value of h for each parallelogram. 1. 2. h 14 For additional support when completing your homework, go to PearsonTEXAS.com. 8 0.5 0.3 3. 13 h 0.4 10 h 12 18 Find the area of each figure. y hsm11gmse_1001_t09165 hsm11gmse_1001_t09162 J K F hsm11gmse_1001_t09156 4 2 A O B 2 4 6 C 8 x D 10 12 4. ▱ ABJF 5.△BDJ 6.△DKJ 7. ▱ BDKJ 8.▱ ADKF 9.△BCJ 10.trapezoid ADJF hsm11gmse_1001_t09177 PearsonTEXAS.com 523 11.Apply Mathematics (1)(A) A bakery has a 50 ft-by-31 ft parking lot. The four parking spaces are congruent parallelograms, the driving region is a rectangle, and the two areas for flowers are congruent triangles. a.Find the area of the paved surface by adding the areas of the driving region and the four parking spaces. 31 ft 10 ft 15 ft b.Describe another method for finding the area of the paved surface. 50 ft c.Use your method from part (b) to find the area. Then compare answers from parts (a) and (b) to check your work. 12.What is the area of the figure at the right? 14 cm A.64 cm2 C.96 cm2 B.88 cm2 D.112 cm2 8 cm 13.The area of a parallelogram is 24 in.2 , and the height is 6 in. Find the length of the corresponding base. HSM11GMSE_1001_a09795 1st pass 12-19-08 Durke 8 cm 14.A right isosceles triangle has area 98 cm2 . Find the length of each leg. 15.Analyze Mathematical Relationships (1)(F) The area of a triangle is 108 in.2 . hsm11gmse_1001_t09175 A base and corresponding height are in the ratio 3 : 2. Find the length of the base and the corresponding height. Find the area of each figure. 16. 17. 25 ft 21 cm 18. 15 cm 25 ft 200 m 120 m 40 m 25 ft 60 m 20 cm Select Techniques to Solve Problems (1)(C) Select a technique (such as mental math, estimation, or number sense) to find the area of the composite figure. Then find the area. 19. hsm11gmse_1001_t09179 hsm11gmse_1001_t09183.ai hsm11gmse_1001_t09181.ai 4 yd 20. 6m 10 yd 6 yd 7m 5m 20 yd For Exercises 21 and 22, (a) graph the lines and (b) find the area of the triangle enclosed by the lines. 21.y = - 12 x + 3, y = 0, x = -2 524 Lesson 13-1 Areas of Parallelograms and Triangles 3 22. y = 4 x - 2, y = -2, x = 4 Find the area of a polygon with the given vertices. 23.E(1, 1), F(4, 5), G(11, 5), H(8, 1) 24. A(3, 9), B(8, 9), C(2, -3), D( -3, -3) 25.D(0, 0), E(2, 4), F(6, 4), G(6, 0) 26.K( -7, -2), L( -7, 6), M(1, 6), N(7, -2) 27.Explain Mathematical Ideas (1)(G) Ki used geometry software to make < > the figure at the right. She< constructed AB and a > point C not on AB . Then she constructed < > line k parallel to AB through point C. Next, Ki constructed point D on line k as well as AD and BD. She dragged point D along line k to manipulate△ABD. How does the area of △ABD change? Explain. C D k B A The Greek mathematician Heron is most famous for this formula for the area of a triangle in terms of the lengths of its sides a, b, and c. A = 1s(s − a)(s − b)(s − c), where s = 12 (a +hsm11gmse_1001_t09427.ai b + c) Use Heron’s Formula and a calculator to find the area of each triangle. Round your answer to the nearest whole number. 28.a = 8 in., b = 9 in., c = 10 in. 29.a = 15 m, b = 17 m, c = 21 m 30.a. Use Heron’s Formula to find the area of this triangle. b. Verify your answer to part (a) by using the formula A = 12 bh. 15 in. 9 in. 12 in. TEXAS Test Practice hsm11gmse_1001_t09184.ai 31.The lengths of the sides of a right triangle are 10 in., 24 in., and 26 in. What is the area of the triangle? A. 116 in.2 B. 120 in.2 C. 130 in.2 D. 156 in.2 32.In quadrilateral ABCD, AB ≅ BC ≅ CD ≅ DA. Which type of quadrilateral could ABCD never be classified as? F. squareG. rectangleH. rhombusJ. kite 33.Are the side lengths of △XYZ possible? Explain. X 4 6 Z Y 11 hsm11gmse_1001_t09429.ai PearsonTEXAS.com 525 13-2 Areas of Trapezoids, Rhombuses, and Kites TEKS FOCUS VOCABULARY TEKS (11)(B) Determine the area of composite twodimensional figures comprised of a combination of triangles, parallelograms, trapezoids, kites, regular polygons, or sectors of circles to solve problems using appropriate units of measure. TEKS (1)(B) Use a problem–solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem–solving process and the reasonableness of the solution. Additional TEKS (1)(A), (1)(F), (6)(D), (9)(B) •Height of a trapezoid – The height of a trapezoid is the perpendicular distance between the bases. •Formulate – create with careful effort and purpose. You can formulate a plan or strategy to solve a problem. •Strategy – a plan or method for solving a problem •Reasonableness – the quality of being within the realm of common sense or sound reasoning. The reasonableness of a solution is whether or not the solution makes sense. ESSENTIAL UNDERSTANDING • You can find the area of a trapezoid when you know its height and the lengths of its bases. • You can find the area of a rhombus or a kite when you know the lengths of its diagonals. Key Concept Area of a Trapezoid The area of a trapezoid is half the product of the height and the sum of the bases. A = 12h(b1 + b2) b1 h b2 Key Concept Area of a Rhombus or a Kite hsm11gmse_1002_t09235.ai The area of a rhombus or a kite is half the product of the lengths of its diagonals. 1 A = 2d1d2 d1 d2 d1 d2 526 Rhombus Kite hsm11gmse_1002_t09243.ai Lesson 13-2 Areas of Trapezoids,hsm11gmse_1002_t14116.ai Rhombuses, and Kites Problem 1 TEKS Process Standard (1)(A) Area of a Trapezoid Which borders of Nevada can you use as the bases of a trapezoid? The two parallel sides of Nevada form the bases of a trapezoid. Geography What is the approximate area of Nevada? A = 12h(b1 + b2) se the formula for area of U a trapezoid. = 12(309)(205 + 511) S ubstitute 309 for h, 205 for b1 , and 511 for b2 . = 110,622 205 mi 309 mi Reno Carson City 511 mi Simplify. Las Vegas The area of Nevada is about 110,600 mi2. Problem 2 hsm11gmse_1002_a09884 Nevada 10p x 13p Finding Area Using a Right Triangle What is the area of trapezoid PQRS? How are the sides related in a 30∙-60∙90∙ triangle? The length of the hypotenuse is 2 times the length of the shorter leg, and the longer leg is 13 times the length of the shorter leg. You can draw an altitude that divides the trapezoid into a rectangle and a 30°-60°-90° triangle. Since the opposite sides of a rectangle are congruent, the longer base of the trapezoid is divided into segments of lengths 2 m and 5 m. h = 213 longer leg = shorter leg A = 12h(b1 + b2) Use the trapezoid area formula. = 12(213)(7 + 5) Substitute 213 for h, 7 for b1, and 5 for b2. = 1213 Simplify. The area of trapezoid PQRS is 1213 m2. P 5m 60 S R Q 7m # 13 S 1st proof 12.15.08 5m R hsm11gmse_1002_t09237.ai h 60 P 2m 5m Q Problem 3 Finding the Area of a Kite Do you need to know the side lengths of the kite to find its area? No. You only need the lengths of the diagonals. hsm11gmse_1002_t09241.ai L What is the area of kite KLMN? Find the lengths of the two diagonals: KM = 2 + 5 = 7m and LN = 3 + 3 = 6m. A = 12d1d2 2m 3m K Use the formula for area of a kite. = 12(7)(6) Substitute 7 for d1 and 6 for d2 . = 21 3m 5m M N Simplify. The area of kite KLMN is 21 m2. hsm11gmse_1002_t09244.ai PearsonTEXAS.com 527 Problem 4 Finding the Area of a Rhombus How can you find the length of AB? AB is a leg of right △ABC. You can use the Pythagorean Theorem, a2 + b2 = c 2, to find its length. Car Pooling The High Occupancy Vehicle (HOV) lane is marked by a series of “diamonds,” or rhombuses painted on the pavement. What is the area of the HOV lane diamond shown at the right? △ABC is a right triangle. Using the Pythagorean Theorem, AB = 26.52 - 2.52 = 6. Since the diagonals of a rhombus bisect each other, the diagonals of the HOV lane diamond are 5 ft and 12 ft. A 6.5 ft B 2.5 ft A = 12 d1d2 Use the formula for area of a rhombus. = 12 (5)(12) Substitute 5 for d1 and 12 for d2 . = 30 Simplify. C The area of the HOV lane diamond is 30 ft2. Problem 5 TEKS Process Standard (1)(B) Finding the Area of a Composite Figure Use a problem-solving model to find the area of the figure below. 6 yd 6 yd 3 yd 3 yd 4 yd 4 yd 10 yd Analyze the Given Information Using the definitions of a kite and a trapezoid, you can determine that this figure is composed of a kite and two trapezoids. The diagonal of the kite is 10 yd long, and the two trapezoids both have base lengths of 6 yd and 4 yd, and a height of 3 yd. Formulate a Plan To find the area of the composite figure, add the areas of each individual figure. continued on next page ▶ 528 Lesson 13-2 Areas of Trapezoids, Rhombuses, and Kites Problem 5 continued Determine and Justify the Solution Find the area of each trapezoid. A = 12 h(b1 + b2) Use the formula for area of a trapezoid. 1 = 2 (3)(6 + 4) Substitute 3 for h, 6 for b1 , and 4 for b2 . = 15 Simplify. Find the area of the kite. The length of the shorter diagonal is 2(6) - 2(4) = 4 yd. A = 12 d1d2 Use the formula for area of a kite. = 12 (4)(10) Substitute 4 for d1 and 10 for d2 . = 20 Simplify. Find the total area. The total area is 15 + 15 + 20 = 50. So the area of the composite figure is 50 yd2. Evaluate the Problem-Solving Process NLINE HO ME RK O What should you do if the answer doesn’t check? You should examine the problem-solving process to find mistakes in your reasoning or your calculations. WO Check your answer. You can divide the composite figure in a different way, find the area, and compare your answers. You can divide the figure into a rectangle and an isosceles triangle. The base of the rectangle is 12 yd and the height is 3 yd, so its area is 36 yd2. The triangle has base 4 yd and height 7 yd, so its area is 14 yd2. 36 + 14 = 50, so the total area is 50 yd2. The answer checks. Since the answer checks, the problem-solving model worked effectively in finding the area of the composite figure. PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. Find the area of quadrilateral QRST. 1. y For additional support when completing your homework, go to PearsonTEXAS.com. R 2. y R 4 Q 3. y R 2 2 x 2 T S -2 2 S 4 x Q O -2 T 2 Q S 2 4 x 2 T 4. The border of Tennessee resembles a trapezoid with bases 340 mi and 440 mi and height 110 mi. Estimate the area of Tennessee by finding the area of the trapezoid. hsm11gmse_1002_t09301.ai hsm11gmse_1002_t09302.ai hsm11gmse_1002_t09300.ai PearsonTEXAS.com 529 Find the area of each trapezoid. If your answer is not an integer, leave it in simplest radical form. 5. 5 ft 6. 3 ft 7. 6m 8 ft 10 m 60 6 ft 8m 15 ft Find the area of each kite. 8. 9. 2hsm11gmse_1002_t09284.ai 10. m 2 in. 3m hsm11gmse_1002_t09282.ai 8 in. 8 in. 8 in. hsm11gmse_1002_t09283.ai 4m 6 ft 3m 4 ft 4 ft Find the area of each rhombus. 11. 12. f 20 hsm11gmse_1002_t09285.ai t 30 ft 13. 10 in. hsm11gmse_1002_t09290.ai 8 in. hsm11gmse_1002_t09288.ai 6m 5m Find the area of each trapezoid. 14. 21 in. 15. hsm11gmse_1002_t09291.ai 16 in. 16. 24.3 cm hsm11gmse_1002_t09293.ai hsm11gmse_1002_t09294.ai 9 ft 6 ft 8.5 cm 18 ft 9.7 cm 38 in. 17.Find the area of a trapezoid with bases 12 cm and 18 cm and height 10 cm. hsm11gmse_1002_t09279.ai 18.Find the area of a trapezoid with bases 2 ft and 3 ft and height 13 ft. hsm11gmse_1002_t09246.ai 19.Use a Problem-Solving Model (1)(B) Find the area of the figure at the right. Use a problemsolving model by • analyzing the given information • formulating a plan or strategy • determining a solution • justifying the solution • evaluating the problem-solving process 530 Lesson 13-2 Areas of Trapezoids, Rhombuses, and Kites hsm11gmse_1002_t09281.ai 26 ft 13 ft 18 ft 24 ft 20.In trapezoid ABCD at the right, AB } DC. Find the area of ABCD. B 15 in. A 20 in. 135 21.Analyze Mathematical Relationships (1)(F) One base of a trapezoid is twice the other. The height is the average of the two bases. The area is 324 cm2. Find the height and the lengths of the bases. 30 D C 22.Apply Mathematics (1)(A) Ty wants to paint one side of the skateboarding ramp hsm11gmse_1002_t09310.ai he built. The ramp is 4 m wide. Its surface is modeled by the equation y = 0.25x2 . Use the trapezoids and triangles shown to estimate the area to be painted. y 1 y 0.25x2 x 2 1 O 1 23.Apply Mathematics (1)(A) The end of a gold bar has the shape of a trapezoid with the measurements shown. Find the area of the end. hsm11gmse_1002_t09309.ai 24.a.Create Representations to Communicate Mathematical Ideas (1)(E) Graph the lines x = 0, x = 6, y = 0, and y = x + 4. 2 6.9 cm 4.4 cm 9.2 cm b.What type of quadrilateral do the lines form? c.Find the area of the quadrilateral. TEXAS Test Practice 25.The area of a kite is 120 cm2. The length of one diagonal is 20 cm. What is the length of the other diagonal? A.12 cm C.24 cm B.20 cm D.48 cm 26.△ABC ∼ △XYZ. AB = 6, BC = 3, and CA = 7. Which of the following are NOT possible dimensions of △XYZ? F. XY = 3, YZ = 1.5, ZX = 3.5 G.XY = 9, YZ = 4.5, ZX = 10.5 H.XY = 10, YZ = 7, ZX = 11 J.XY = 18, YZ = 9, ZX = 21 27.Draw an angle. Construct a congruent angle and its bisector. PearsonTEXAS.com 531 13-3 Areas of Regular Polygons TEKS FOCUS VOCABULARY TEKS (11)(A) Apply the formula for the area of regular polygons to solve problems using appropriate units of measure. •Apothem – An apothem is the perpendicular distance from the center of a polygon to a side. •Center of a regular polygon – The center of a regular polygon is the center of a circle circumscribed about the polygon. TEKS (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. Additional TEKS (1)(D), (9)(B) •Radius of a regular polygon – A radius of a regular polygon is the distance from the center of the polygon to a vertex. •Analyze – closely examine objects, ideas, or relationships to learn more about their nature ESSENTIAL UNDERSTANDING The area of a regular polygon is related to the distance from the center to a side. Postulate 13-2 If two figures are congruent, then their areas are equal. Key Concept Area of a Regular Polygon The area of a regular polygon is half the product of the apothem and the perimeter. A = 12ap a p hsm11gmse_1003_t09350.ai 532 Lesson 13-3 Areas of Regular Polygons Problem 1 TEKS Process Standard (1)(F) Finding Angle Measures The figure at the right is a regular pentagon with radii and an apothem drawn. What is the measure of each numbered angle? How do you know the radii make isosceles triangles? Since the pentagon is a regular polygon, the radii are congruent. So the triangle made by two adjacent radii and a side of the polygon is an isosceles triangle. 360 m∠1 = 5 = 72 Divide 360 by the number of sides. m∠2 = 12m∠1 T he apothem bisects the vertex angle of the isosceles triangle formed by the radii. 90 + 36 + m∠3 = 180 The sum of the measures of the angles of a triangle is 180. hsm11gmse_1003_t09347.ai m∠3 = 54 2 1 = 12(72) = 36 3 m∠1 = 72, m∠2 = 36, and m∠3 = 54. Problem 2 Finding the Area of a Regular Polygon What is the area of the regular decagon shown below? 12.3 in. What do you know about the regular decagon? A decagon has 10 sides, so n = 10. From the diagram, you know that the apothem a is 12.3 in., and the side length s is 8 in. 8 in. Step 1 Find the perimeter of the regular decagon. p = ns = 10(8) = 80 in. Use the formula for the perimeter of an n-gon. hsm11gmse_1003_t09351.ai Substitute 10 for n and 8 for s. Step 2 Find the area of the regular decagon. A = 12ap = 12(12.3)(80) = 492 Use the formula for the area of a regular polygon. Substitute 12.3 for a and 80 for p. The regular decagon has an area of 492 in.2. PearsonTEXAS.com 533 Problem 3 TEKS Process Standard (1)(D) Using Special Triangles to Find Area STEM Zoology A honeycomb is made up of regular hexagonal cells. The length of a side of a cell is 3 mm. What is the area of a cell? You know the length of a side, which you can use to find the perimeter. The apothem Draw a diagram to help find the apothem. Then use the area formula for a regular polygon. Step 1 Find the apothem. The radii form six 60° angles at the center, so you can use a 30°-60°-90° triangle to find the apothem. a = 1.513 longer leg = 13 # shorter leg Step 2 Find the perimeter. p = ns Use the formula for the perimeter of an n-gon. Substitute 6 for n and 3 for s. = 6(3) = 18 mm 30 534 3 mm hsm11gmse_1003_t09353.ai A = 12ap Use the formula for the area of a regular polygon. = 12(1.513) (18) Substitute 1.513 for a and 18 for p. ≈ 23.3826859 Lesson 13-3 Areas of Regular Polygons 60 1.5 mm Step 3 Find the area. The area is about 23 mm2. a Use a calculator. Problem 4 Finding the Area of a Composite Figure The figure below is composed of two congruent regular hexagons and two triangles. What is the area of the figure? Round your answer to the nearest square meter. 9m Step 1 Find the area of one of the regular hexagons. To find the area of a regular polygon, you need to know the apothem and the perimeter. Use a 30°-60°-90° triangle to find the apothem. Since the hypotenuse is 9 m long, the length of the apothem is 4.513 m. The perimeter of the hexagon is 6 9 m, or 54 m. # A = 12ap Use the formula for the area of a regular polygon. = 12(4.513)(54) Substitute 4.5 13 for a and 54 for p. = 121.5 13Simplify. Step 2 Find the area of one of the triangles. The measure of each angle of a regular hexagon is 120. So the measure of an exterior angle is 180 - 120 = 60. Since two exterior angles of the hexagons make up two angles of the triangle, the measure of all angles of the triangle must be 60. Therefore, it is an equilateral triangle with side length 9 m. Use another 30°-60°-90° triangle to find the height. Since the hypotenuse is 9 m, the height is 4.513 m. A = 12bh Use the formula for the area of a triangle. 1 = 2(9)(4.5 13) Substitute 9 for b and 4.5 13 for h. = 20.2513Simplify. How do you know that the two triangles are congruent? The triangles are both equilateral, with side lengths of 9 m. So they are congruent. Step 3 Find the area of the four figures combined. A = 121.513 + 121.513 + 20.2513 + 20.2513 = 283.513 Simplify. ≈ 491.0364039 Use a calculator. The area of the composite figure is about 491 m2. PearsonTEXAS.com 535 HO ME RK O NLINE WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. Each regular polygon has radii and apothem as shown. Find the measure of each numbered angle. For additional support when completing your homework, go to PearsonTEXAS.com. 1. 2. 3. 4 1 2 7 5 8 6 3 9 Find the area of each regular polygon with the given apothem a and side length s. 4. pentagon, a = 24.3 cm, s = 35.3 cm hsm11gmse_1003_t09363.ai hsm11gmse_1003_t09364.ai hsm11gmse_1003_t09367.ai 5. octagon, a = 60.4 in., s = 50 in. 6. nonagon, a = 27.5 in., s = 20 in. 7. dodecagon, a = 26.1 cm, s = 14 cm Find the area of each regular polygon. Round your answer to the nearest tenth. 8. 18 ft 9. 10. 8 in. 6m 11.Use Multiple Representations to Communicate Mathematical Ideas (1)(D) You are painting a mural of colored equilateral triangles. The radius of each triangle is 12.7 in. What is the hsm11gmse_1003_t09369.ai area of each triangle to the nearest square inch? hsm11gmse_1003_t09368.ai hsm11gmse_1003_t09370.ai s 2 30 12.7 in. s Find the area of each regular polygon with the given radius or apothem. If your answer is not an integer, leave it in simplest radical form. 13.hsm11gmse_1003_t09371.ai 14. 12. 4 in. 6 cm 536 8 V3 in. hsm11gmse_1003_t09373.ai hsm11gmse_1003_t09376.ai hsm11gmse_1003_t09372.ai Lesson 13-3 Areas of Regular Polygons 15.Apply Mathematics (1)(A) The gazebo in the photo is built in the shape of a regular octagon. Each side is 8 ft long, and the enclosed area is 310.4 ft2. What is the length of the apothem? STEM 16.Apply Mathematics (1)(A) One of the smallest space satellites ever developed has the shape of a pyramid. Each of the four faces of the pyramid is an equilateral triangle with sides about 13 cm long. What is the area of one equilateral triangular face of the satellite? Round your answer to the nearest whole number. 17.A regular hexagon has perimeter 120 m. Find its area. 18.The area of a regular polygon is 36 in.2. Find the length of a side if the polygon has the given number of sides. Round your answer to the nearest tenth. a.3 b.4 c.6 d.Select Techniques to Solve Problems (1)(C) Suppose the polygon is a pentagon. What would you expect the length of a side to be? Explain. 19.A portion of a regular decagon has radii and an apothem drawn. Find the measure of each numbered angle. 2 3 1 20.Explain Mathematical Ideas (1)(G) Explain why the radius of a regular polygon is greater than the apothem. Find the area of each composite figure. Assume that all parts of figures shown hsm11gmse_1003_t09377.ai are regular polygons and that figures that are the same shape are congruent. Leave your answer in simplest radical form. 22 in. 21.22. 23. 14 m 8 ft Find the perimeter and area of each regular polygon. Round to the nearest tenth, as necessary. 24.a square with vertices at ( -1, 0), (2, 3), (5, 0), and (2, -3) 25.a hexagon with two adjacent vertices at ( -2, 1) and (1, 2) PearsonTEXAS.com 537 26.To find the area of an equilateral triangle, you can use the formula A = 12bh or A = 12ap. A third way to find the area of an equilateral triangle is to use the formula A = 14s2 13. Verify the formula A = 14s2 13 in two ways, as follows: a.Find the area of Figure 1 using the formula A = 12bh. b.Find the area of Figure 2 using the formula A = 12ap. s s s 2 s 2 Figure 1 27.For Problem 1, write a proof showing that the apothem Proof bisects the vertex angle of an isosceles triangle formed by two radii. Figure 2 28.Prove that the bisectors of the angles of a regular polygon are concurrent and that Proof they are, in fact, radii of the polygon. (Hint: For regular n-gon hsm11gmse_1003_t09381.ai ABCDE . . ., let P be the intersection of the bisectors of ∠ABC and ∠BCD. Show that DP must be the bisector of ∠CDE.) y 29.Analyze Mathematical Relationships (1)(F) A regular octagon with center at the origin and radius 4 is graphed in the coordinate plane. a.Since V2 lies on the line y = x, its x- and y-coordinates are equal. Use the Distance Formula to find the coordinates of V2 to the nearest tenth. V2 2 2 O 2 V1 (4, 0) x 2 b.Use the coordinates of V2 and the formula A = 12bh to find the area of △V1OV2 to the nearest tenth. c.Use your answer to part (b) to find the area of the octagon to the nearest whole number. hsm11gmse_1003_t09382.ai TEXAS Test Practice 30.What is the area of a regular pentagon with an apothem of 25.1 mm and a perimeter of 182 mm? A.913.6 mm2 B.2284.1 mm2 C.3654.6 mm2 D.4568.2 mm2 31.What is the most precise name for a regular polygon with four right angles? F. square G.parallelogram H.trapezoid J.rectangle 32.△ABC has coordinates A( -2, 4), B(3, 1), and C(0, -2). If you reflect △ABC across the x-axis, what are the coordinates of the vertices of the image △A′B′C′? A.A′(2, 4), B′( -3, 1), C′(0, -2) C. A′(4, -2), B′(1, 3), C′( -2, 0) B.A′( -2, -4), B′(3, -1), C′(0, 2) D. A′(4, 2), B′(1, -3), C′( -2, 0) 33.An equilateral triangle on a coordinate grid has vertices at (0, 0) and (4, 0). What are the possible locations of the third vertex? Explain. 538 Lesson 13-3 Areas of Regular Polygons 13-4 Perimeters and Areas of Similar Figures TEKS FOCUS VOCABULARY •Justify – explain with logical reasoning. TEKS (10)(B) Determine and describe how changes in the linear dimensions of a shape affect its perimeter, area, surface area, or volume, including proportional and non-proportional dimensional change. You can justify a mathematical argument. •Argument – a set of statements put TEKS (1)(G) Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication. forth to show the truth or falsehood of a mathematical claim Additional TEKS (1)(F), (11)(A) ESSENTIAL UNDERSTANDING You can use ratios to compare the perimeters and areas of similar figures. Key Concept Changes in Dimension Examples A proportional dimensional change multiplies every dimension by the same value. Under a proportional dimensional change, the image of a figure is similar to its preimage. Under a nonproportional dimensional change, the image of a figure is not similar to its preimage. Two examples of nonproportional dimensional changes are the following: • Each dimension has the same constant value added to it • Each dimension is multiplied by a different value a ∙ 1.5 a b ∙ 1.5 b a+2 b+2 a b a∙2 b∙3 a b Key Concept Perimeters and Areas of Similar Figures a If the scale factor of two similar figures is b , then a • the ratio of their perimeters is b • the ratio of their areas is a2 b2 PearsonTEXAS.com 539 Problem 1 TEKS Process Standard (1)(F) Analyzing Proportional Dimensional Changes How does multiplying each dimension of the A 5 cm isosceles trapezoid by a scale factor of 2 affect its perimeter? How does multiplying each dimension by a scale factor of 3 affect its perimeter? 5 cm 4 cm 5 cm 11 cm Find the perimeter of the trapezoid. P = 5 + 11 + 5 + 5 = 26 Find the dimensions and the perimeter of each scaled trapezoid. Compare the new perimeter to the original perimeter of 26 cm. Find the new dimensions. Scale Factor 2 Scale Factor 3 # Find the new perimeter. # # b1 = 2 5 = 10 b2 = 2 11 = 22 leg = 2 5 = 10 h = 2 4 = 8 # # b1 = 3 5 = 15 b2 = 3 11 = 33 leg = 3 5 = 15 h = 3 4 = 12 # # # P = 10 + 22 + 10 + 10P = 15 + 33 + 15 + 15 = 52 = 78 52 cm = 2 # 26 cm 78 cm = 3 # 26 cm So, when each dimension is multiplied by 2, the perimeter is multiplied by 2. When each dimension is multiplied by 3, the perimeter is multiplied by 3. B How does multiplying each dimension of the isosceles trapezoid in Part A by a scale factor of 2 affect its area? How does multiplying each dimension by a scale factor of 3 affect its area? Find the area of the trapezoid. How does the ratio of the areas appear to be related to the scale factors? Since 2 2 = 4 and 3 3 = 9, the ratio of the areas appears to be the square of the scale factors. # 540 # A = 12(4)(5 + 11) = 32 Use the dimensions you calculated in Part A to find the area of each scaled trapezoid. Compare the new area to the original area of 32 cm2. Scale Factor 2 A = 12 h(b1 + b2) A= = 12(8)(10 + 22) = = 128 128 cm2 = 4 # 32 cm2 Scale Factor 3 1 2 h(b1 + b2) 1 2(12)(15 + 33) = 288 288 cm2 = 9 # 32 cm2 So, when each dimension is multiplied by 2, the area is multiplied by 4. When each dimension is multiplied by 3, the area is multiplied by 9. Lesson 13-4 Perimeters and Areas of Similar Figures Problem 2 Finding Ratios in Similar Figures How do you find the scale factor? Write the ratio of the lengths of two corresponding sides. The trapezoids at the right are similar. The ratio of the lengths of corresponding sides is 69 , or 23 . 6m A What is the ratio (smaller to larger) of the perimeters? The ratio of the perimeters is the same as the ratio of corresponding sides, which is 23 . 9m B What is the ratio (smaller to larger) of the areas? The ratio of the areas is the square of the ratio of corresponding 2 sides, which is 22 , or 49 . 3 Problem 3 hsm11gmse_1004_t09322.ai Finding Areas Using Similar Figures Can you eliminate any answer choices immediately? Yes. Since the area of the smaller pentagon is 27.5 cm2, you know that the area of the larger pentagon must be greater than that, so you can eliminate choice A. Multiple Choice The area of the smaller regular pentagon is about 27.5 cm2. What is the best approximation for the area of the larger regular pentagon? 11 cm2 69 cm2 172 cm2 4 cm 10 cm 275 cm2 Regular pentagons are similar because all angles measure 108 and all sides in each pentagon are congruent. Here the ratio of corresponding side lengths 2 hsm11gmse_1004_t09323.ai 4 4 is 10 , or 25 . The ratio of the areas is 22 , or 25 . 5 27.5 4 25 = A Write a proportion using the ratio of the areas. 4A = 687.5 Cross Products Property 687.5 A= 4 Divide each side by 4. A = 171.875 Simplify. The area of the larger pentagon is about 172 cm2. The correct answer is C. Problem 4 TEKS Process Standard (1)(G) Applying Area Ratios Do you need to know the shapes of the two plots of land? No. As long as the plots are similar, you can compare their areas using their scale factor. Agriculture During the summer, a group of high school students cultivated a plot of city land and harvested 13 bushels of vegetables that they donated to a food pantry. Next summer, the city will let them use a larger, similar plot of land. In the new plot, each dimension is 2.5 times the corresponding dimension of the original plot. How many bushels can the students expect to harvest next year? The ratio of the dimensions is 2.5 : 1. So the ratio of the areas is (2.5)2 : 12, or 6.25 : 1. With 6.25 times as much land next year, the students can expect to harvest 6.25(13), or about 81, bushels. PearsonTEXAS.com 541 Problem 5 Finding Perimeter Ratios The triangles shown below are similar. What is the scale factor? What is the ratio of their perimeters? Area 50 cm2 The areas of the two similar triangles a2 50 = b2 98 a2 25 = b2 49 a 5 = b 7 Area 98 cm2 The scale factor hsm11gmse_1004_t15486 Write a proportion using the ratios of the areas. Use a2 : b2 for the ratio of the areas. Simplify. Take the positive square root of each side. The ratio of the perimeters equals the scale factor 5 : 7. Problem 6 Analyzing Nonproportional Dimension Changes Are the rectangles similar? No. Since you cannot apply the same scale factor to the lengths of each side of one of the rectangles to get the other, they are not similar. The botany club plans to increase the size of a rectangular garden by adding 8 ft to each dimension of the garden. 4 ft A The botany club wants to put fencing around the proposed garden. How many more feet of fencing will the club need to buy for the proposed garden than it would have bought for the current garden? Find the perimeter of the current garden. 22 ft 4 ft P = 2 # 22 + 2 # 16 = 76 Find the perimeter of the proposed garden. P = 2(22 + 8) + 2(16 + 8) = 108 16 ft Find the difference of the two perimeters. 108 - 76 = 32 The botany club will need to buy 32 more feet of fencing for the proposed garden. continued on next page ▶ 542 Lesson 13-4 Perimeters and Areas of Similar Figures Problem 6 continued B The botany club wants to cover the proposed garden with a layer of mulch. How much greater is the area of the proposed garden than the area of the current garden? Find the area of the current garden. A = 22 # 16 = 352 Find the area of the proposed garden. A = (22 + 8)(16 + 8) = 720 Find the difference of the two areas. 720 - 352 = 368 NLINE HO ME RK O So the area of the proposed garden is 368 yd2 greater than the area of the current garden. WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. The figures in each pair are similar. Compare the first figure to the second. Give the ratio of the perimeters and the ratio of the areas. For additional support when completing your homework, go to PearsonTEXAS.com. 1. 2 in. 2. 3. 15 in. 4 in. 8 cm 25 in. 6 cm Find the scale factor and the ratio of perimeters for each pair of similar figures. 4. two regular octagons with areas 4 ft2 and 16 ft2 hsm11gmse_1004_t09327.ai hsm11gmse_1004_t09325.ai 5. two trapezoids with areas 49 hsm11gmse_1004_t09330.ai cm2 and 9 cm2 6. two equilateral triangles with areas 1613 ft2 and 13 ft2 7. two circles with areas 2p cm2 and 200p cm2 Analyze Mathematical Relationships (1)(F) Find the values of x and y when the smaller triangle shown here has the given area. 8 cm x y 12 cm 8. 3 cm2 9.6 cm2 10.12 cm2 11.16 cm2 12.24 cm2 13.48 cm2 hsm11gmse_1004_t09335.ai PearsonTEXAS.com 543 The figures in each pair are similar. The area of one figure is given. Find the area of the other figure to the nearest whole number. 14. 15. 3 in. 12 m 6 in. Area of smaller parallelogram = 6 in.2 16. Area of larger trapezoid = 121 m2 17. hsm11gmse_1004_t09331.ai 16 ft 18 m hsm11gmse_1004_t09332.ai 12 ft 3m 11 m Area of larger triangle = 105 ft2 Area of smaller hexagon = 23 m2 hsm11gmse_1004_t09333.ai 18.Apply Mathematics (1)(A) An embroidered placemat costs $3.95. An embroidered tablecloth is similar to the placemat, but four times as long and hsm11gmse_1004_t09334.ai four times as wide. How much would you expect to pay for the tablecloth? 19.The longer sides of a parallelogram are 5 m. The longer sides of a similar parallelogram are 15 m. The area of the smaller parallelogram is 28 m2. What is the area of the larger parallelogram? STEM 20.Apply Mathematics (1)(A) For some medical imaging, the scale of the image is 3 : 1. That means that if an image is 3 cm long, the corresponding length on the person’s body is 1 cm. Find the actual area of a lesion if its image has area 2.7 cm2. 21.A rectangular pool and its scale drawing are similar, with a scale factor of 2.5 in. : 11.5 ft. If the dimensions of the drawing are 5.5 in. by 11 in., what is the area of the bottom of the actual pool? 22.A rectangular driveway has a perimeter of 56 feet. If the length is increased by 4 feet, how is the perimeter affected? What is the new perimeter? 23.A postcard has side lengths s and t. Determine the changes in the area and perimeter of the postcard if the length of s is tripled. 24.Explain Mathematical Ideas (1)(G) A reporter used the graphic below to show that the number of houses with more than two televisions had doubled in the past few years. Explain why this graphic is misleading. Then 544 Now Lesson 13-4 Perimeters and Areas of Similar Figures STEM 25.a.Create Representations to Communicate Mathematical Ideas (1)(E) A surveyor measured one side and two angles of a field, as shown in the diagram. Use a ruler and a protractor to draw a similar triangle. b.Measure the sides and altitude of your triangle and find its perimeter and area. 50 30 200 yd c.Estimate the perimeter and area of the field. 26.Suppose the lengths of both bases of the isosceles trapezoid are halved. Describe how the area of the trapezoid is affected. 27.a.Find the area of a regular hexagon with sides 2 cm long. Leave your answer in simplest radical form. b.Use your answer to part (a) and ratios to find the areas of the regular hexagons shown at the right. 28.Justify Mathematical Arguments (1)(G) The enrollment at an elementary school is going to increase from 200 students to 395 students. A parents’ group is planning to increase the 100 ft-by-200 ft playground area to a larger area that is 200 ft by 400 ft. What would you tell the parents’ group when they ask your opinion about whether the new playground will be large enough? 6 cm 7 cm 4 cm HSM11GMSE_1004_a09815 12 cm 1st pass 12-19-08 Durke 3 cm 8 cm hsm11gmse_1004_t09340.ai 29.The figure at the right is a scale drawing of a patio. The scale of the drawing is 2 cm = 5 ft. What is the perimeter of the actual patio? 12 cm 30.A 3 in.-by-5 in. photograph is enlarged by a scale factor of 1 in. : 1.5 ft. 8 cm a.Find the perimeter and the area of the enlarged photograph. 8 cm b.Suppose the length and width of the enlarged photograph are doubled. Describe how the perimeter and area are affected. c.Suppose the length (the measure of the longer side) of the enlarged photograph is doubled. Describe how the perimeter and area are affected. TEXAS Test Practice 31.What is the value of x in the diagram at the right? 32.Two regular hexagons have sides in the ratio 3 : 5. The area of the smaller hexagon is 81 m2. In square meters, what is the area of the larger hexagon? 21 26 7 x 33.A trapezoid has base lengths of 9 in. and 4 in. and a height of 3 in. What is the area of the trapezoid in square inches? 34.In quadrilateral ABCD, m∠A = 62, m∠B = 101, and m∠C = 42. What is m∠D? hsm11gmse_1004_t09536.ai PearsonTEXAS.com 545 13-5 Trigonometry and Area TEKS FOCUS VOCABULARY TEKS (9)(A) Determine the lengths of sides and measures of angles in a right triangle by applying the trigonometric ratios sine, cosine, and tangent to solve problems. TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. •Number sense – the understanding of what numbers mean and how they are related Additional TEKS (1)(F), (11)(A) ESSENTIAL UNDERSTANDING • You can use trigonometry to find the area of a regular polygon • You can use trigonometry to find the area of a triangle if if you know the length of a side, a radius, or an apothem. you know the lengths of two sides and the included angle. Theorem 13-1 Area of a Triangle Given SAS B The area of a triangle is half the product of the lengths of two sides and the sine of the included angle. Area of △ABC = 12 bc (sin A) a c A b C For a proof of Theorem 13-1, see the Reference section on page 683. Problem 1 TEKS Process Standard (1)(F) hsm11gmse_1005_t10016.ai Finding Area What is the area of a regular nonagon with 10-cm sides? What is the apothem in the diagram? The apothem is the altitude of the isosceles triangle. The apothem bisects the central angle and the side of the polygon. Draw a regular nonagon with center C. Draw CP and CR to form isosceles △PCR. The measure of central ∠PCR is 360 9 , or 40. The perimeter is 9 10, or 90 cm. Draw the apothem CS. # 10 cm C m∠PCS = 12 m∠PCR = 20 and PS = 12 PR = 5 cm Let a represent CS. Find a and substitute into the area formula. 5 tan 20° = a Use the tangent ratio. 5 a= tan 20° A = 12 ap = 12 tan520° # # 90 ≈ 618.1824194 Solve for a. 5 Substitute for a and 90 for p. tan 20° Use a calculator. P S C a R 20 hsm11gmse_1005_t10011.ai P 5 S The area of the regular nonagon is about 618 cm2. 546 Lesson 13-5 Trigonometry and Area hsm11gmse_1005_t10012.ai Problem 2 TEKS Process Standard (1)(C) Finding Area Road Signs A stop sign is a regular octagon. The standard size has a 16.2-in. radius. What is the area of the stop sign to the nearest square inch? The radius and the number of sides of the octagon Use trigonometric ratios to find the apothem and the length of a side The apothem and the length of a side Step 1Let a represent the apothem. Use the cosine ratio to find a. The measure of a central angle of the octagon is 360 8 , or 45. So m∠C = 12 (45) = 22.5. a cos 22.5° = 16.2 16.2(cos 22.5°) = a C Use the cosine ratio. 22.5 Multiply each side by 16.2. 16.2 a Step 2Let x represent AD. Use the sine ratio to find x. x sin 22.5° = 16.2 16.2(sin 22.5°) = x Multiply each side by 16.2. Step 3 Find the perimeter of the octagon. The length of each side is 2x. # length of a side = 8 # 2x = 8 # 2 # 16.2(sin 22.5°) = 259.2(sin 22.5°) Simplify. A x D Use the sine ratio. hsm11gmse_1005_t10013.ai p=8 Substitute for x. 7 4 2 . . . . . . . Step 4 Substitute into the area formula. A = 12ap = 12 16.2(cos 22.5°) # # 259.2(sin 22.5°) ≈ 742.2924146 The area of the stop sign is about 742 in.2. Substitute for a and p. Use a calculator. 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 PearsonTEXAS.com 547 Problem 3 Finding Area Which formula should you use? The diagram gives the lengths of two sides and the measure of the included angle. Use the formula for the area of a triangle given SAS. What is the area of the triangle? Area = 12 = 12 # side length # side length # sine of included angle # 12 # 21 # sin 48° Substitute. ≈ 93.63624801 12 cm 48 21 cm Use a calculator. The area of the triangle is about 94 cm2. HO ME RK O hsm11gmse_1005_t10017.ai NLINE WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. Select Tools to Solve Problems (1)(C) Find the area of each regular polygon. Round your answers to the nearest tenth. For additional support when completing your homework, go to PearsonTEXAS.com. 1. octagon with side length 6 cm 2.decagon with side length 4 yd 3. pentagon with radius 3 ft 4.nonagon with radius 7 in. 5. dodecagon with radius 20 cm 6.20-gon with radius 2 mm 7. 18-gon with perimeter 72 mm 8.15-gon with perimeter 180 cm Find the area of each triangle. Round your answers to the nearest tenth. 9. 11 m 12. 57 10. 6m 34 km 11. 104 m 12 ft 1 2 33 40 5 ft 226 m 13. 11 mm 1 ft 37 14. 28 76 hsm11gmse_1005_t10024.ai 1 hsm11gmse_1005_t10025.ai hsm11gmse_1005_t10023.ai 2 ft 39 km 24 mm 2 15.Analyze Mathematical Relationships (1)(F) PQRST is a regular pentagon with center O and radius 10 in. Find each measure. If necessary, round your answers to the nearest tenth. hsm11gmse_1005_t10026.ai hsm11gmse_1005_t10029.ai hsm11gmse_1005_t10028.ai P T X S O Q R a.m∠POQ b.m∠POX c.OX d.PQ e.perimeter of PQRST 548 Lesson 13-5 Trigonometry and Area f.area of PQRST hsm11gmse_1005_t10027.ai 16.Apply Mathematics (1)(A) The Pentagon in Arlington, Virginia, is one of the world’s largest office buildings. It is a regular pentagon, and the length of each of its sides is 921 ft. What is the area of land that the Pentagon covers to the nearest thousand square feet? 17.Apply Mathematics (1)(A) The surveyed lengths of two adjacent sides of a triangular plot of land are 80 yd and 150 yd. The angle between the sides is 67°. What is the area of the parcel of land to the nearest square yard? Find the perimeter and area of each regular polygon to the nearest tenth. 18. 19. 4m 3 ft 20. 21. 1 mi hsm11gmse_1005_t10030.ai 10 m hsm11gmse_1005_t10031.ai 22.What is the area of the triangle shown below? hsm11gmse_1005_t10032.ai 53 10 cm hsm11gmse_1005_t10033.ai 79 8 cm 23.The central angle of a regular polygon is 10°. The perimeter of the polygon is 108 cm. What is the area of the polygon? hsm11gmse_1005_t10034.ai PearsonTEXAS.com 549 Regular polygons A and B are similar. Compare their areas. 24.The apothem of Pentagon A equals the radius of Pentagon B. 25.The length of a side of Hexagon A equals the radius of Hexagon B. 26.The radius of Octagon A equals the apothem of Octagon B. 27.The perimeter of Decagon A equals the length of a side of Decagon B. 28.Replacement glass for energy-efficient windows costs $5/ft2. About how much will you pay for replacement glass for a regular hexagonal window with a radius of 2 ft? A.$10.39 B.$27.78 C.$45.98 D.$51.96 The polygons are regular polygons. Find the area of the shaded region. 29. 30. 8 cm 31. 6 ft 6 ft 4 in. 6 cm 32.Segments are drawn between the midpoints of consecutive sides of a regular pentagon to form another regular pentagon. Find, to the nearest hundredth, the hsm11gmse_1005_t10036.ai hsm11gmse_1005_t10035.ai ratio of the area of the smaller pentagon to the area of the larger pentagon. hsm11gmse_1005_t10037.ai STEM 33.Apply Mathematics (1)(A) A surveyor wants to mark off a triangular parcel with an area of 1 acre (1 acre = 43,560 ft2). One side of the triangle extends 300 ft along a straight road. A second side extends at an angle of 65° from one end of the first side. What is the length of the second side to the nearest foot? TEXAS Test Practice 34.A regular polygon has a perimeter of 54 m and an apothem of 313 m. What is the area of the polygon to the nearest tenth of a square meter? 35.The legs of a right triangle have lengths of 8 in. and 15 in. What is the length of the hypotenuse in inches? 36.△PEN ≅ △LIV . If m∠P = 36 and m∠N = 82, what is m∠I ? 37.The perimeter of a parallelogram is 23.6 ft. If its length and width are doubled, what is the perimeter of the parallelogram in feet? 38.The altitude to the hypotenuse of a right triangle divides the hypotenuse into segments of lengths 8 and 10. What is the length of the shorter leg of the triangle? 550 Lesson 13-5 Trigonometry and Area Topic 13 Review TOPIC VOCABULARY • altitude of a parallelogram, p. 520 • center of a regular polygon, p. 532 • height of a trapezoid, p. 526 • apothem, p. 532 • composite figure, p. 520 • height of a triangle, p. 520 • base of a parallelogram, p. 520 • height of a parallelogram, p. 520 • radius of a regular polygon, p. 532 • base of a triangle, p. 520 Check Your Understanding Choose the vocabulary term that correctly completes the sentence. 1.A ? is a combination of two or more figures. 2.An ? is the perpendicular distance from the center of a polygon to a side. 3.The distance from the center to a vertex is the ? . 4.The length of the altitude of a parallelogram is also called the ? . 13-1 Areas of Parallelograms and Triangles Quick Review Exercises You can find the area of a rectangle, a parallelogram, or a triangle if you know the base b and the height h. The area of a rectangle or parallelogram is A = bh. Find the area of each figure. h 5. b 1 The area of a triangle is A = 2 bh. 6 ft What is area of the parallelogram? Use the area formula. = (12)(8) = 96 Substitute and simplify. The area of the parallelogram is 96 cm2. 10 in. 9 in. 4m hsm11gmse_10cr_t10605.ai 7. Example A = bh 6. 5m 12 cm 8 cm 8. 10 ft hsm11gmse_10cr_t10610.ai 16 ft 10 ft hsm11gmse_10cr_t10609.ai 9.A right triangle has legs measuring 5 ft and 12 ft, and hypotenuse measuring 13 ft. What is its area? hsm11gmse_10cr_t10638 .ai hsm11gmse_10cr_t10611.ai hsm11gmse_10cr_t10637.ai PearsonTEXAS.com 551 13-2 Areas of Trapezoids, Rhombuses, and Kites Quick Review Exercises The area of a trapezoid is A = 12 h(b1 + b2). The area of a rhombus or a kite is d1 A = 12 d1d2 , where d1 and d2 are the lengths of its diagonals. Find the area of each figure. If necessary, leave your answer in simplest radical form. b2 The height of a trapezoid h is the perpendicular distance between the bases, b1 and b2. h 10. b1 11 mm 8 ft 8 ft d2 60 6 hsm11gmse_10cr_t10606.aimm 12. 10 ft 15 mm 13. 6.5 cm 10 8 cm hsm11gmse_10cr_t10613.ai ft hsm11gmse_10cr_t10612.ai .8 12 ft Example What is the area of the trapezoid? A = 12 h(b1 + b2) = 12 (8)(7 + 3) = 40 11. Use the area formula. 3 cm hsm11gmse_10cr_t10607.ai 10 cm 8 cm 6.5 cm Substitute. Simplify. 7 cm 14. A trapezoid has a height of 6 m. The length of one base is three times the length of the other base. The hsm11gmse_10cr_t10640 .ai sum of the base lengths is 18 m. What is the area of the hsm11gmse_10cr_t10614.ai trapezoid? The area of the trapezoid is 40 cm2. hsm11gmse_10cr_t10639 .ai 13-3 Areas of Regular Polygons Quick Review Exercises The center of a regular polygon C is the center of its circumscribed circle. The radius r is the distance from the center to a vertex. The apothem a is the perpendicular distance from the center to a side. The area of a regular polygon with apothem a and perimeter p is A = 12 ap. Find the area of each regular polygon. If your answer is not an integer, leave it in simplest radical form. C r a 15. 6 in. 16. 7 m p 17. What is the area of a regular hexagon with a perimeter of 240 cm? Example hsm11gmse_10cr_t10608.aihsm11gmse_10cr_t10641 .ai 18. What is the area of a square withhsm11gmse_10cr_t10642 radius 7.5 m? .ai What is the area of a hexagon with apothem 17.3 mm and perimeter 120 mm? Sketch each regular polygon with the given radius. Then A = 12 ap Use the area formula. find its area to the nearest tenth. = 12(17.3)(120) = 1038 Substitute and simplify. The area of the hexagon is 1038 mm2. 19. triangle; radius 4 in. 20. square; radius 8 mm 21. hexagon; radius 7 cm 552 Topic 13 Review 13-4 Perimeters and Areas of Similar Figures Exercises Quick Review a If the scale factor of two similar figures is b , then the ratio of a2 a their perimeters is b , and the ratio of their areas is 2 . b For each pair of similar figures, find the ratio of the area of the first figure to the area of the second. 22. 8 12 23. Example If the ratio of the areas of two similar figures is 49 , what is the ratio of their perimeters? 6 Find the scale factor. 14 2 = 19 3 24. Take the square root of the ratio of areas. 4 25. 3 hsm11gmse_10cr_t10615.ai 6 7 mm 14 mm hsm11gmse_10cr_t10616.ai The ratio of the perimeters is the same as the ratio of corresponding sides, 23 . 26. If the ratio of the areas of two similar hexagons is 8 : 25, what is the ratio of their perimeters? hsm11gmse_10cr_t10631.ai hsm11gmse_10cr_t10617.ai 13-5 Trigonometry and Area Quick Review Exercises You can use trigonometry to find the areas of regular polygons. You can also use trigonometry to find the area of a triangle when you know the lengths of two sides and the measure of the included angle. Find the area of each polygon. Round your answers to the nearest tenth. Area of a △ = 12 side length # 27. regular decagon with radius 5 ft 28. regular pentagon with apothem 8 cm # side length # sine of included angle 29. regular hexagon with apothem 6 in. 30. regular quadrilateral with radius 2 m Example 31. regular octagon with apothem 10 ft What is the area of △XYZ? Area = 12 = 12 X # XY # XZ # sin X # 15 # 13 # sin 65° ≈ 88.36500924 The area of △XYZ is approximately 88 ft2. 15 ft Y 65 32. regular heptagon with radius 3 ft 13 ft Z 33. 34. 15 cm 12 m 45 19 cm 78 12 m hsm11gmse_10cr_t10618.ai hsm11gmse_10cr_t10619.ai hsm11gmse_10cr_t10620.ai PearsonTEXAS.com 553 Topic 13 TEKS Cumulative Practice Multiple Choice 5.Every triangle in the figure at the right is an equilateral triangle. What is the total area of the shaded triangles? Read each question. Then write the letter of the correct answer on your paper. s2 13 s2 13 s 6.Which of the following could be the side lengths of a right triangle? B. 25 m2D. 513 m2 2.If △CAT is rotated 90° around vertex C, what are the coordinates of A′? 4 F. 4.1, 6.2, 7.3H. 3.2, 5.4, 6.2 y C G. 40, 60, 72J. 33, 56, 65 2 F. (1, 5) T A O 2 2 x 4 2 H. ( -1, 3) hsm11gmse_10cu_t10743.ai 7.The vertices of △ART are A(1, -2), R(5, -1), and T(4, -4). If T65, -27 (△ART) = A′R′T′, what are the coordinates of the image triangle A′R′T′? A. A′(5, -1), R′(4, -4), T′(1, -2) B. A′(6, -4), R′(10, -3), T′(9, -6) J. (3, 5) 3.Which transformation can you use to justify that △ABC is congruent to △PQR? A 4 2 D. A′( -1, 3), R′(3, 4), T′(2, 1) B R 6 ft x 4 4 2 2 P C. A′(3, 3), R′(7, 4), T′(6, 1) Bhsm11gmse_10cu_t10740.ai 8.In the figure below, what is the length of AD? y O C 4 Q A. a reflection across the y-axis B. a reflection across the x-axis C. a rotation 180° clockwise around the origin hsm11gmse_10cu_t10741.ai D. a translation 5 units left and 6 units down 4.Your friend is 5 ft 6 in. tall. When your friend’s shadow is 6 ft long, the shadow of a nearby sculpture is 30 ft long. What is the height of the sculpture to the nearest tenth? F. 32.7 ftH. 27.5 ft G. 28 ftJ. 25 ft 554 s2 13 B.12 D. 6 A. 2513 m2 C. 1013 m2 G. (3, 3) s2 13 A.36 C. 9 1.What is the exact area of an equilateral triangle with sides of length 10 m? Topic 13 TEKS Cumulative Practice A D 4 ft C F. 2 113 ftH. 10 ft G. 9 ftJ. 3 113 ft hsm11gmse_10cu_t10745.ai 9.For what value of x are the two triangles similar? 5x 2 2 3 3x 2 6 4 1 A. 4 2 C. B. 2D. 6 hsm11gmse_09cu_t08650.ai Gridded Response 10. Find the area, in square meters, of the figure. 18. A photographic negative is 3 cm by 2 cm. A similar print from the negative is 9 cm long on its shorter side. What is the length, in centimeters, of the longer side? 19. What is the geometric mean of 10 and 15? 5m 3m 8m Constructed Response 5m 11. Find the area, in square millimeters, of the regular polygon. Round to the nearest tenth. 10 mm 12. A triangle has a perimeter of 81 in. If you divide the length of each side by 3, what is the perimeter, in inches, of the new triangle? 13. The diagonal of a rectangular patio makes a 70° angle with a side of the patio that is 60 ft long. What is the area, to the nearest square foot, of the patio? 14. What is the area, in square feet, of the unshaded part of the rectangle below? 250 ft 120 ft 50 ft 100 ft 15. You are making a scale model of a building. The front of the actual building is 60 ft wide and 100 ft tall. The front of your model is 3 ft wide and 5 ft tall. What is the scale factor ofhsm11gmse_10cu_t10747.ai the reduction? 16. A square has an area of 225 cm2. If you double the length of each side, what is the area, in square centimeters, of the new square? 17. A meter stick perpendicular to the ground casts a 1.5-m shadow. At the same time, a telephone pole casts a shadow that is 9 m. How tall, in meters, is the telephone pole? 20. The coordinates of △ABC are A(2, 3), B(10, 9), and C(10, -3). Is △ABC an equilateral triangle? Explain. 21. Draw a right triangle. Then construct a second triangle congruent to the first. Show your steps. 22. One diagonal of a parallelogram has endpoints at P( -2, 5) and R(1, -1). The other diagonal has endpoints at Q(1, 5) and S( -2, -1). What type of parallelogram is PQRS? Explain. 23. A right triangle has height 7 cm and base 4 cm. Find its area using the formulas A = 12 bh and A = 12 ab(sin C). Are the results the same? Explain. 24. Can a regular polygon have an apothem and a radius of the same length? Explain. 25. The coordinates of the vertices of isosceles trapezoid ABCD are A(0, 0), B(6, 8), C(10, 8), and D(16, 0). The coordinates of the vertices of isosceles trapezoid AFGH are A(0, 0), F(3, 4), G(5, 4), and H(8, 0). Are the two trapezoids similar? Justify your answer. 26. At a campground, the 50-yd path from your campsite to the information center forms a right angle with the path from the information center to the lake. The information center is located 30 yd from the bathhouse. How far is your campsite from the lake? Show your work. Information center 50 yd Your campsite 30 yd x Bathhouse y Lake hsm11gmse_07cu_t05666 PearsonTEXAS.com 555