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Transcript
Section 3.2 Triangles Pre-Activity Preparation Geena needs to make sure that the deck she is building is perfectly square to the brace holding the deck in place. How can she use geometry to ensure that the boards are aligned properly? Geena can use the Pythagorean Theorem (see information below). Every right triangle has special properties related to the lengths of its sides, its angles, and even the relationship of its angles to its sides. The mathematical field of trigonometry is the study of right triangle relationships. Each deck board must make a square corner with the brace. A square corner is a right angle. Geena can use the Pythagorean Theorem to ensure that the corner is a right angle by measuring the two sides of a triangle formed by the brace and board. She will measure 4″ along one side and make a point, and 3″ along the other side and make a point. If the line drawn connecting the two points measures exactly 5″, then the angle opposite that line is a right angle. 5” 4” 3” Learning Objectives • Learn how to apply the Pythagorean Theorem to find the length of a side of a right triangle, given its other two sides • Use the rules for congruent triangles to determine if two triangles are congruent • Find the perimeter and area of a triangle given its base and height Terminology Previously Used New Terms to Learn acute angle area isosceles angle base legs obtuse angle congruent triangles perimeter perpendicular corresponding sides Pythagorean Theorem square root equilateral right triangle vertex height scalene hypotenuse side 173 Chapter 3 — Geometry 174 Building Mathematical Language Triangles Three distinct non-parallel lines, lying in the same plane, will intersect at three distinct points and form a triangle. Triangles are named based upon these points of intersection. The triangle to the right is called “triangle ABC” or “ABC.” (While triangle names are usually alphabetized—“ABC” rather than “CAB”— this is not always the case.) a B b A c C • The interior angles of the triangle are a, b, and c. The sum of the interior angles of a triangle is always 180°: ∠a + ∠b + ∠c = 180° • The base or bottom of triangle ABC is the segment BC, AB, or AC depending on the orientation of the triangle. (Orientation means the way the triangle is turned). • The height (also called the altitude) is the perpendicular distance from the top point to the base. Find the height by measuring the length of a line drawn from the top vertex that intersects the base at a right angle. • Each pair of intersecting lines forms four angles—two pairs of vertical angles. • Each interior angle is adjacent to two supplementary exterior angles and across from one vertical exterior angle. Types of Triangles Name Picture Special Features Observations No sides are equal and no angles are equal Scalene triangles are the “generic” triangle and have no special features. Two equal sides and two equal angles Usually presented with the base and equal angles at the bottom. The equal sides are opposite the equal angles. Equilateral All angles are equal and all sides are equal An equilateral triangle has angles that measure 60°. Acute All three angles are acute Each angle measures less than 90°. An equilateral triangle is acute. One angle is obtuse One obtuse angle; therefore, two acute angles. Scalene Isosceles Obtuse Section 3.2 — Triangles 175 Name Picture Special Features Observations Right One angle is a right angle. The side opposite the right angle is the hypotenuse. The other two sides can be called legs. Many concepts in mathematics and many real world applications use right triangles. Isosceles Right The legs of the right angle are equal and the two acute angles are equal and measure 45°. Dividing a square on one of its diagonals creates two isosceles right triangles. Dividing a square on both its diagonals yields four isoceles right triangles. Perimeter of a Triangle Perimeter is the distance around an object in a plane. Perimeter is a measure of length and, therefore, measured in feet (ft), inches (in), meters (m), centimeters (cm), etc. In this context, the Greek peri- means around and -meter means measure, so the word “perimeter” means measure around. The perimeter of a triangle is the sum of the measures of the sides: Perimeter P=a+b+c B The perimeter of triangle ABC P=a+b+c P = 4.9 + 7.8 + 12 P = 24.7 c = 12 A b = 7.8 a = 4.9 C Area of a Triangle Area measures the amount of surface of an object in a plane. Measure area in terms of square feet (ft2), square meters (m2), square miles (mi2), etc. The formula for finding the area of a triangle is one-half times the base times the height: Area A = ½bh B Area = ½ base × height The area of right triangle ABC A = ½bh A = (½)(12m)(4m) A = (6 × 4)m2 A = 24m2 height = 4m Validate: 24m2 ÷ 4m = 6m; 6m ÷ 12m = ½ C base (b) = 12m A Chapter 3 — Geometry 176 Congruent Triangles Triangles that are the same size and shape are congruent triangles. Congruent triangles have equal corresponding angles and equal corresponding side lengths. Use the symbol “,” to indicate congruency. A b=7 C a=5 D e=7 c = 3.5 B F f = 3.5 d=5 E ∠A , ∠D and a,d ∠B , ∠E and b,e ∠C , ∠F and c,f Triangle ABC , DEF Techniques Determining Congruency between Triangles To determine if two triangles are congruent, match their corresponding parts and then use one of the following rules. Types of Triangle Congruency Name S-S-S side–side–side S-A-S side-angle-side Description Two triangles are congruent if corresponding sides are equal. Two sides and the enclosed angle of one triangle are equal to the corresponding two sides and enclosed angle of the other triangle. Two angles and the enclosed side of one triangle are equal to the corresponding angle-side-angle two angles and enclosed side of the other triangle. Visual The tick marks indicate which sides are conguent. Observations If you can determine that corresponding side measures are equal, then the triangles are congruent. The equal angle must be between the two equal sides. The tick marks indicate conguent sides and angles. A-S-A The tick marks indicate conguent angles and sides. The equal side must be between the two equal angles. Section 3.2 — Triangles 177 Pythagorean Theorem The Pythagorean Theorem is the most widely used and important relationship in geometry. Knowledge of this theory has existed for millennia, with the first recorded statement found on a Babylonian tablet (c. 1600 B.C.). While various sources contain proofs of the relationship (some of which certainly predate Pythagoras’ work), the theorem was named after him, as it was his work that became most widely known. Pythagorean Theorem In any right triangle ABC, where c is the side opposite the right angle: B c a a2 + b2 = c2 (Note: side a is across from angle A and side b is across from angle B.) C b A Example: Given right triangle ABC, and a = 3, b = 4, and c = 5: a2 + b2 = c2 32 + 42 = 52 9 + 16 = 25 25 = 25 B 3 C 5 4 A Write down the Pythagorean Theorem for the triangle below. Label the right angle as angle C and c = 10, b = 8, and a = 6. Try it! a2 + b2 = c2 B a 2 a C Area of Square a + Area of Square b = Area of Square c c2 You can easily learn more about the Pythagorean Theorem online. c Here are some fun sites to get you started: A b http://www.cut-the-knot.org/pythagoras/ http://www.mathsnet.net/dynamic/pythagoras/ b 2 http://mathworld.wolfram.com/PythagoreanTheorem.html http://www.pbs.org/wgbh/nova/proof/puzzle/theorem.html a2 + b2 = c2 Chapter 3 — Geometry 178 Methodology Using the Pythagorean Theorem: a2 + b2 = c2 Label each triangle with C as its right angle. ► Example 1: Find c if a = 7 and b = 12. Round to the nearest tenth, if necessary. C ► Example 2: Find c if a = 11 and b = 15. Round to the nearest tenth, if necessary. Steps in the Methodology Step 1 Identify a, b and c on the given triangle. Any of the three sides can be the unknown value. Make a sketch if necessary. Substitute the given side lengths into the formula. Example 1 c=? a=7 C b = 12 a=7 b =12 c = unknown C Step 2 Try It! If c is not the unknown side, use the equivalent formula to find side a or b: a2 + b2 = c2 72 + 12 2 = c2 c2 – b2 = a2 or c – a2 = b2 2 Step 3 Solve for the square of the unknown side using order of operations. Step 4 Use your calculator to find the positive square root. Step 5 Validate Solve for a2 or b2 if you are using the alternative formulas. Use your calculator if necessary. 49 + 144 = c2 Round off to the desired number of places. If you round off, use ≈ not =. 193 = c 2 Use the original information to check for equality. 72 + 12 2 = 13.9 2 193 = c2 193 = c 13.9 . c ? ? 49 + 144 = 193.2 193 ≈ 193.2 Example 2 Section 3.2 — Triangles 179 Models B Model 1 Find the perimeter and area of the given right triangle. c = 24.5 in a = h = 14.7 in C Perimeter P=a+b+c A b = 19.6 in Validate: 58.8 in – 24.5 in = 34.3 in P = 14.7 + 19.6 + 24.5 34.3 in – 19.6 in = 14.7 in P = 58.8 inches Area A = ½bh Validate: A = ½(19.6)(14.7) A = (9.8)(14.7) A = 144.06 square inches (144.06 in 2 ) = 9.8 in 14.7 in (9.8 in ) = (9.8 in ) × 2 = 19.6 in 1 2 B Model 2 Find the perimeter and area of the given triangle. a = 50 in c = 41 in h = 40 in C Perimeter determine side b (base): 30 in 9 in A Validate: 130 in – 41 in = 89 in 30 in + 9 in = 39 in 89 in – 50 in = 39 in P=a+b+c P = 50 + 39 + 41 P = 130 inches Area A = ½bh A = ½(39)(40) A = (19.5)(40) A = 780 square inches Validate: (780 in 2 ) = 19.5 in 40 in (19.5 in ) = (19.5 in ) × 2 = 39 in 1 2 Chapter 3 — Geometry 180 Model 3 Given a right triangle, find side b if side a = 3.9 and side c (hypotenuse) = 6.5. Approximate the answer to the nearest tenth, if necessary. Step 1 Identify a = 3.9 c = 6.5 a = 3.9 b = unknown b c = 6.5 Step 2 Substitute c2 – a2 = b2 (Alternative formula to find side b.) 6.52 – 3.92 = b2 Step 3 Solve 42.25 – 15.21 = b2 27.04 = b2 Step 4 Square root 5.2 = b Step 5 Validate 5.22 + 3.92 = 6.52 ? ? 27.04 + 15.21 = 42.25 42.25 = 42.25 Model 4 Given a right triangle, find side a if side b = 11 and side c (hypotenuse) = 18.2. Approximate the answer to the nearest tenth, if necessary. a Step 1 Identify c = 18.2 a = unknown b = 11 c = 18.2 Step 2 Substitute c2 – b2 = a2 b = 11 18.22 – 112 = a2 Step 3 Solve 331.24 – 121 = a2 210.24 = a2 Step 4 Square root 14.5 ≈ a Step 5 Validate 14.52 + 112 = 18.22 ? ? 210.25 + 121 = 331.24 331.25 ≈ 331.24 Section 3.2 — Triangles 181 Model 5a B Are the triangles in the pair at right congruent? Use what you know about triangles, including types of congruency and how to solve for unknown sides in right triangles, in order to determine if the triangles at right are congruent with each other. Provide an explanation for your answer(s). E 45˚ a 45˚ 1 C c d = 3m b = 3m A F The missing angle in triangle 1 (∠A) is 45°: ∠A = 180° – 90° – 45° = 45° Triangle 2 is also an isosceles triangle and side e = 3m. With the enclosed angle of 90°, the S-A-S rule of congruity is satisfied. See the diagram below. E 45˚ a = 3m C = 90° ANGLE SIDE 1 45˚ b = 3m A ANGLE 45˚ e = 3m D SIDE 2 We can also apply the Pythagorean Theorem to find the length of the hypotenuse, to show S-S-S congruency. a = 3m, b = 3m c2 = 32 + 32 = 18 c ≈ 4.24 m d = 3m, e = 3m f 2 = 32 + 32 = 18 f ≈ 4.24 m B E 45˚ a = 3m SIDE 1 C 45˚ c = 4.24m SIDE 3 d = 3m 1 45˚ b = 3m SIDE 1 A SIDE 2 F f = 4.24m 2 SIDE 3 45˚ e = 3m A-S-A Angle 1: ∠C , ∠F Side: b,e Angle 2: ∠A , ∠D Knowing the missing hypotenuse for each triangle also enables us to show S-A-S congruency in two additional ways: S-S-S Side 1: a , d Side 2: b , e Side 3: c , f 1 , 2 D SIDE 2 Once we have used the Pythagorean Theorem to find sides c and f, we have all angle and side measurements for each triangle. This allows us to show A-S-A congruency, in three different ways: A-S-A Angle 1: ∠B , ∠E Side: a,d Angle 2: ∠C , ∠F 1 , 2 2 F = 90° SIDE 2 Side 1: a , d Angle: ∠C , ∠F Side 2: b , e 45˚ d = 3m 1 45˚ e S-A-S Triangle 1 is therefore an isosceles triangle and side a = 3m. SIDE 1 2 Type of Congruency Reason B f A-S-A 1 , 2 A-S-A Angle 1: ∠A , ∠D Side: c,f Angle 2: ∠B , ∠E S-A-S Side 1: a , d Angle: ∠B , ∠E Side 2: c , f S-A-S Side 1: c , f Angle: ∠A , ∠D Side 2: b , e D Chapter 3 — Geometry 182 Model 5b Are the triangles in the pair at right congruent? Use what you know about triangles, including types of congruency and how to solve for unknown sides in right triangles. Provide an explanation for your answer. 1.8 48˚ 80˚ 52˚ 1 2 x 52˚ 1.8 2.3 Type of Congruency Reason These are not right triangles, so we cannot use the Pythagorean Theorem to find missing side lengths. However, we already have equivalent sides of 1.8 and equivalent angles of 52°. If we can solve for the missing angle (∠x) in triangle 2 and it is equal to 80°, then we can show A-S-A congruency. ∠x = 180° – 52° – 48° = 80°. The 80° angles, the 52° angles, and their enclosed sides of 1.8 satisfy the A-S-A rule of congruency. See the diagram below. A-S-A Angle 1: 80° = 80° Side: 1.8 = 1.8 Angle 2: 52° = 52° 1 , 2 ANGLE 1 SIDE 1.8 ANGLE 2 80˚ 48˚ 52˚ 1 2.3 2 ANGLE 1 80˚ 52˚ 1.8 SIDE ANGLE 2 Model 5c Determine whether or not the triangles in the pair at right congruent. Use what you know about triangles, including types of congruency and how to solve for unknown sides in right triangles. Provide an explanation for your answer. 12.3 99° 8.7 99° 12.3 21.6 Reason Keep in mind that you cannot determine congruency by “looks.” You must use S-S-S, S-A-S, or A-S-A to show congruency. The largest corresponding angles equal 99°, so they are not right triangles and the Pythagorean Theorem cannot be used. While the triangle on the right has the given angle enclosed with the two given sides, the triangle on the left does not; therefore, the relationship does not fit any of the three rules for congruency. Type of Congruency NOT CONGRUENT Section 3.2 — Triangles 183 Addressing Common Errors Incorrect Process Issue Misidentifying corresponding sides or angles If triangle ABC is congruent to triangle XYZ, Orientation matters. Rearrange the triangles so that the largest angles are matched. (9 ABC / 9XYZ) which sides are equal measures? c a C z A b C Z y X Y c a x b If side a = 12 and side b = 6, what is the measure of side c? a z x A Z y X a=x b=y c=z a=z c=x b=y Use of the Pythagorean theorem on non-right triangles Validation The second triangle should be rotated or flipped until angle Z is in the same position as angle C. B Y B Correct Process Resolution The Pythagorean Theorem is only applicable to RIGHT triangles. b c c 2 = a 2 + b 2 , so Careful observation reveals that the triangle is not a right triangle; no indication of a right angle is present. The Pythagorean Theorem is not useful here. Side c cannot be determined with the information given. c 2 = 122 + 62 and c 2 = 144 + 36 c 2 = 180 c = 180 . 13.4 Applying the Pythagorean Theorem incorrectly In right triangle ABC: A c b C a if a = 5 and b = 7 find c. a+b=c 5 + 7 = 12 = c B The Pythagorean Theorem states that the SQUARES of the two legs added together equals the SQUARE of the hypotenuse. ? a2 + b2 = c2 52 + 72 = 8.62 52 + 72 = c2 25 + 49 = 73.96 25 + 49 = c2 74 = c2 c = 74 . 8.6 ? 74 ≈ 73.96 Chapter 3 — Geometry 184 Incorrect Process Issue Using the wrong side as the hypotenuse in the Pythagorean formula Triangle RST is a right triangle. Find side s if r = 3 and t=5 The hypotenuse is always opposite the right angle. T s r S Correct Process Resolution Validation In the given figure, the right angle is opposite side t, not side s. Use is the alternative formula for finding side s: s2 = t2 – r2. R t t2 = r2 + s2 ? 52 = 32 + 42 ? 25 = 9 + 16 25 = 25 s2 = t2 – r2 s2 = r 2 + t 2 s2 = 52 – 32 s 2 = 9 + 25 s2 = 16 s2 = 25 – 9 s 2 = 32 + 52 s=4 s 2 = 34 s = 34 s . 5.8 Not using square units for area Find the area of a triangle whose base is 10 mm and height is 6mm. A = ½bh A = (½)(6)(10) Area is always given in square units. Carry the units along with the measures to remind you to report the units as part of the answer. A = ½bh (mm)(mm) = mm2 A = (½)(6mm)(10mm) A = 30 mm2 A = 30 mm Not using an appropriate piece of information in a formula. Find the area of the given triangle: A c B b a C When a = 6, b = 10, and h = 8. A = ½bh A = (½)(6)(10) A = 30 square units Make sure that you correctly identify each piece of information before using it in a formula. Clearly labeling a drawing or sketch can help. A b = 10 c h=8 B a=6 C From the new drawing, the height is clearly shown. A = ½bh A = (½)(6)(8) A = 24 square units Using the area calculated, solve for h: A = ½bh 24 = (½)(6)(h) 24 = 3h 24 =h 3 h = 8, the height as given. Section 3.2 — Triangles 185 Preparation Inventory Before proceeding, you should be able to: Find the area and perimeter of a right triangle Find the perimeter and area of a triangle, given its base and height. Use the Pythagorean Theorem to find the unknown side of a right triangle Find the area and perimeter of triangles using the Pythagorean Theorem Use the congruency rules to establish the congruency of two triangles CAN a Triangle Have Three 90-Degree Angles? Of course not, but it is a fun optical illusion. It was first created by the Swedish artist Oscar Reutersvärd in 1934, but later independently devised and made popular by the mathematician Roger Penrose. He described it as “impossibility in its purest form.” You can learn more by searching the internet for the “Penrose triangle.” Impossible Triangle sculpture by Brian MacKay & Ahmad Abas. It is located at Claisebrook roundabout, East Perth, Western Australia. This Penrose triangle only appears to have three 90-degree angles when seen from certain perspectives. Section 3.2 Activity Triangles Performance Criteria • Finding the third side of a right triangle when given the measures of two of its sides – appropriate use of the Pythagorean Theorem to find the third side – correct calculation of the measure of the third side • Calculating perimeters and areas of triangles – correct and appropriate use of the perimeter formula for a triangle – correct and appropriate use of the area formula for a triangle • Calculating the measures of sides and angles in two given triangles – correct identification of corresponding parts – correct calculation of missing angle or side measures – correct determination of congruency Critical Thinking Questions 1. If two triangles are congruent by the S-S-S rule, are the triangles always equilateral triangles? Explain. 2. If you know that two triangles have two pairs of equal corresponding angles, what can you determine about the third pair of angles? 3. Why can’t “angle-angle-angle” be a rule to determine two congruent triangles? 186 Section 3.2 — Triangles 187 4. Can a triangle have two obtuse angles? Explain your answer. 5. Why can you eliminate the negative square root when finding the square root in the Pythagorean Theorem? 6. Why is the height needed to measure the area of a triangle? 7. Given a triangle with the following side lengths: side a = 3 feet side b = 4 meters side c = 2 yards, can you calculate the perimeter? If so, what units would your answer have? Explain your reasoning. Chapter 3 — Geometry 188 Tips for Success • If a figure is not provided, make a quick sketch showing all measurements • Use graph paper to sketch more accurately • Make sure the measurements are in the same units Demonstrate Your Understanding 1. Find the perimeter of the following triangles: Problem Worked Solution a) 22.5 in 18 in 13.5 in b) 18.5 m 6.8 m 6m 20.7 m c) 12 m 15 m 9m d) 17.3 cm 10.8 cm 9.6 cm Validation Section 3.2 — Triangles 189 2. Find the area of the following triangles: Problem Worked Solution a) 18 in 22.5 in 13.5 in b) 18.5 m 6m 6.8 m 20.7 m c) A right triangular tract of land has side a = 41.6 miles and side b = 31.2 miles. 31.2 mi 41.6 mi d) 11 in 12 in Validation Chapter 3 — Geometry 190 3. Use the Pythagorean Theorem to solve the following problems. Round your answers to the nearest tenth, if necessary. Problem a) Worked Solution Find c. c 11 in 12 in b) Find b. 20 in 15 in b c) Find a. a c = 19.4 b = 13 d) Find c and e. c=? a = 7.5 h=4 e=? d = 4.2 Validation Section 3.2 — Triangles 191 Problem Worked Solution e) Find the perimeter of the largest triangle in problem 3 d). f) Find the area of the largest triangle in problem 3 d). Validation 4. Are the triangles in the following pairs congruent? Use what you know about triangles, including types of congruency and how to solve for unknown sides in right triangles, in order to determine if the triangles are congruent with each other. Provide an explanation for your answer(s). Triangle Pair Reason a) 3.9 in 1 c 5.2 in 6.5 in 2 d 5.2 in Type(s) of Congruency Chapter 3 — Geometry 192 Triangle Pair Reason b) 7 cm A 60˚ 7 cm 1 7 cm c) B 27˚ e 60˚ 7 cm 60˚ 2 f x 33˚ w 120˚ 27˚ E Type(s) of Congruency Section 3.2 — Triangles Identify and 193 Correct the Errors In the second column, identify the error(s) in the worked solution or validate its answer. If the worked solution is incorrect, solve the problem correctly in the third column and validate your answer. Identify Errors or Validate Worked Solution 1) The two right triangles below are congruent. If ∠1 = 43°, what is the measure of ∠2? 2 1 ∠1 = ∠2 (corresponding parts) so ∠2 = ∠43° 2) Find the measure of side b if c = 22 in and a = 8 in. B A C c2 = b2 + a2 82 = b2 + 222 64 = b2 + 484 b2 = 420 b ≈ 20.5 in 3) Find the measure of side a if c = 2.5 in and b = 1.5 in. B C A a2 = c2 – b2 a2 = 2.52 – 1.52 a = 6.25 – 2.25 a = 4 in Correct Process Validation Chapter 3 — Geometry 194 Identify Errors or Validate Worked Solution 4) Find the area of triangle DEF. D e = 8.5 f=5 E h=4 d = 10.5 F A = ½bh A = (½)(10.5)(5) A = 26.25 square units 5) A right triangle has ∠1 = 26°. What is the measure of ∠2? 1 2 The triangle is a right triangle, with one angle = 90°. The three angles must add to 180°, therefore: 180° – (90° + 26°) = 180° – 116° = 64° ∠2 = 64° Correct Process Validation