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Name: ______ Period _________ 2/17/12 – 3/2/12 G-PreAP UNIT 12: SIMILARITY I can define, identify and illustrate the following terms: Dilation Scale Factor Extremes Means Similarity Statement Scale Drawing Enlargement Reduction Ratio Proportion Cross products Indirect measurement Similarity ratio Dates, assignments, and quizzes subject to change without advance notice. Monday 13 12- 4 and 12 – 5 Compositions and Symmetry 20 Tuesday Review 12-7 Dilations 28 7-6 Dilations & Similarity in the Coordinate Plane 7-1 Ratios, Proportions, and Problem Solving 24 7-3 & 7-4 Triangle Similarity & Applications QUIZ 1 7-2 Similar Polygons No School Friday 17 TEST 11 22/23 21 27 Block Day 15/16 14 7-5 Proportional Relationships 29/1 8-1 Geometric Mean QUIZ 2 Review 2 TEST 12 Friday, 2/17 7-1: Ratio and Proportion I can write a ratio. I can write a ratio expressing the slope of a line. I can solve a linear proportion. I can solve a quadratic proportion. PRACTICE: Pg 458 #8-13, 17-29 Tuesday, 2/21 7-2: Similar Polygons I can use the definition of similar polygons to determine if two polygons are similar. I can determine the similarity ratio between two polygons. PRACTICE: Pg 465 #7-20, 27-30 Ratio Problems Worksheet Wednesday, 2/22 or Thursday, 2/23 QUIZ 1: 7-1 & 7-2 7-3: Triangle Similarity & 7-4: Applications and Problem Solving I can use the triangle similarity theorems to determine if two triangles are similar. I can use proportions in similar triangles to solve for missing sides. I can use the triangle proportionality theorem and its converse. I can use the Triangle Angle Bisector Theorem. I can set up and solve problems using properties of similar triangles. PRACTICE: Pg 475 #11-14, 16, 20-24 Pg 485 #8-20, 25, 28, 32, 34 Friday, 2/24 7-5: Using Proportional Relationships I can use proportions to determine if a figure has been dilated. I can use ratios to make indirect measurements. PRACTICE: Pg 491 #18-19, 24-26, 28-29, 34 Proportional Relationships Worksheet Monday, 2/27 7-6: Dilations & Similarity in the Coordinate Plane I can use coordinate proof to prove figures similar. I can apply similarity in the coordinate plan. PRACTICE: Pg 498 #4-7, 11-14, 21-24 Tuesday, 2/28 QUIZ 2: Similar Triangles (7-3 through 7-6) Geometric Mean I can use geometric mean. PRACTICE: Geometric Mean Worksheet Wednesday, 2/29 or Thursday, 3/1 Review Test 12: Similarity I can use ratios and proportions to show figures are similar or solve problems with similar figures. PRACTICE: Review Worksheet; Pg 504 – 507 has even MORE practice if you want it. Dilations as Proportions Notes Ex) Rectangle CUTE was dilated to create rectangle UGLY. Find the length of LY. C 8 cm 3 cm E U T U 7.5 cm Y G L Ex) Determine which of the following figures could be a dilation of the triangle to the right. (There could be more than one answer) A 16 in. C B 20 in. D 8 in. 6 in. 30 in. 2.25 in. 6 in. 3 in. 10 in. 5 in. 1. Find the length of AB after the dilation. A′ A 4m 6m C 10 m B C′ B′ 2. Which of the following could NOT be an enlargement or reduction (dilation) of the original painting shown at right? 8 in 12 in A 4 in B 10 in 6 in 15 in C D 12 in 9in 13 in 18 in Ratio Problems Write the equation for each and solve. Show all work. 1. One angle of a triangle measures 18º. The other two angles are in the ratio of 4:5. Find the measures of the angles. 2. The vertex angle and the one base angle of an isosceles triangle are in the ratio of 7:4. Find the measures of the 3 angles. 3. The sides of a triangle have the ratio of 7:9:12. The perimeter of the triangle is 420. Find the length of each side. 4. Two consecutive angles of a parallelogram are in a ratio of 7:3. Find the measures of the angles. 5. The angles of a pentagon are in a ratio of 7:6:5:5:4. Find the measures of each angle. 6. One angle of a hexagon is 120º. The other angles are in the ratio of 11:9:8:7:5. Find the measures of the angles. 7. The sides of a quadrilateral are 5, 7, 4, and 8 cm. If the shortest side of a similar quadrilateral is 10cm, what is the length of the longest side? 8. If a rectangular I.D. card is 4cm x 8cm (l x w) is enlarged so that its new perimeter is 50cm, what is the new width? 9. The base of an isosceles triangle is 12cm long and one of the legs is 6cm long. What is the perimeter of a similar triangle whose base is 18cm? 10. The sides of a polygon are 3, 4, 7, 9, and 11 mm long. Find the perimeter of a similar polygon whose shortest side is 10.5 mm long. Notes: Similar Triangles There are 3 ways you can prove triangles similar WITHOUT having to use all sides and angles. Angle- Angle Similarity (AA~) – If two angle of one triangle are ______________ to two corresponding angles of another triangle, then the triangles are similar Side- Side- Side Similarity (SSS~) – If the three sides of one triangle are __________________ to the three corresponding sides of another triangle, then the triangles are similar. Side-Angle- Side Similarity (SAS~) – If two sides of one triangle are ____________________ to two corresponding sides of another triangle and their included angles are ________________, then the triangles are similar. Examples: Determine if the triangles are similar. If so, tell why and write the similarity statement and similarity ratio. Similar : Y or N Why:_________ Similar : Y or N Why:_________ Similarity Statement :_______~__________ Similarity Statement :_______~__________ Similarity Ratio :__________ Similarity Ratio :__________ Similar : Y or N Similar : Y or N Why:_________ Why:_________ Similarity Statement :_______~__________ Similarity Statement :_______~__________ Similarity Ratio :__________ Similarity Ratio :__________ Notes: Properties of Similar Triangles Ex. 1 Ex. 2 D I V Y ID DE IV E O ID DE DV A E L R P S AY YE LA AY AE YR LY PL RS PA AY ES Now that you can write the proportions, you can solve problems. D Ex. 3 Ex. 5 Ex. 4 5 I x x Ex. 6 Find RV V x Ex. 7 Find y E 8 20 O Proportional Relationships Solve for the variable in each figure. 1. y =__________ 10 x = __________ 2. k =______________ 5y – 2 5 12 3x + 2 2 3 9 2 k 4 3 x 3. SQ = x ; ST = 22 ; SP = 12 ; PR = 4x+8 4. y =__________ x = __________ x 8 x = __________ 16 5 y 3 20 Given: j k l A j D B k E C F 5. AC = 9; BC = 6; DF = 15 EF = __________ 6. AB = 5y; DE = 2y; EF = 12 BC = ___________ l 7. BC = x + 2; BA = 9; EF = x + 3; ED = 12 x = _______; BC = ________; EF = _______ 8. AC = 3x; BC = 16; EF = 20; FD = 4x -2 x = ______; AC = _______; FD = _______ Dilations & Similarity in the Coordinate Plane Notes 5. 7. 6. 8. 9. 10. 11. Given: E(–2, –6), F(–3, –2), G(2, –2), H(–4, 2), and J(6, 2). Prove: ∆EHJ ~ ∆EFG. 12. Given: R(–2, 0), S(–3, 1), T(0, 1), U(–5, 3), and V(4, 3). Prove: ∆RST ~ ∆RUV Notes: Geometric Mean The geometric mean of two positive numbers is the positive square root of their products. For a x , the x x b is the geometric mean. Examples: Find the geometric mean of the given numbers. a. 4 and 9 b. 6 and 15 c. 2 and 8 The altitude to the hypotenuse of a right triangle forms two triangles that are ________________ to each other and to the original triangle. In other words, there are ________ similar triangles: a small one, a medium one, and a large one. Examples: Ex 1: m = 2, n = 10, h = _____ a b h m n c c h a b a m b n h Ex 2: m = 2, n = 10, b = _____ a You draw the similar triangles: b h m n c Geometric Mean If ∆ABC is a right triangle and CD is the altitude to the hypotenuse AB then C m a →a² = mc ∆ABC ~ ∆CBD → a c n b ∆ABC ~ ∆ACD → → b² = nc a b c h m h ∆ACD ~ ∆CBD → → h² = mn h n A m D n B c c a b h b a m b n h 1. c = 12; m = 6; a = ? 2. m = 4; h = 25; n = ? 3. c = 12; m = 4; h = ? 4. a = 30; c = 50; h = ? 5. h = 12; m = 9; b = ? 6. a = 24; m = 4; b = ? 7. b = 45; n = 5; a = ? 8. b = 8; m = 12; c =? 10. a = 7 5 ; h = 14; c = ?; n = ? 11. a = 6 5 ; b = 3 5 ; m = ?; h = ?