8.1 Day 1 Warmup. Solve each equation. 1. 4x + 5x + 6x = (x 5) 2 = 81. in simplest form. 3. Write 16
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1 8.1 Day 1 Warmup Solve each equation. 1. 4x + 5x + 6x = (x 5) 2 = Write in simplest form. 4. If QRS ZYX, identify the pairs of congruent angles and the pairs of congruent sides. February 9, 2017 Geometry 8.1 Similar Polygons 1
2 8.1 Day 2 Warmup Solve each proportion. 1. x 4 = x 9 = 4 x 3. 4 x 12 = = 8 x 3 3x = x 3 x+1 5 February 9, 2017 Geometry 8.1 Similar Polygons 2
3 Geometry 8.1 Similar Polygons
4 8.1 Essential Question How are similar polygons related? February 9, 2017 Geometry 8.1 Similar Polygons 4
5 Goals Solve proportions. Identify similar polygons Find the ratio of similarity between similar figures. Solve problems involving similar figures. February 9, 2017 Geometry 8.1 Similar Polygons 5
6 Ratio Is a common fraction. A comparison of two numbers by division. The denominator cannot be zero. The ratio of a to b can be written: a : b or a b February 9, 2017 Geometry 8.1 Similar Polygons 6
7 Simplifying Ratios Ratios must be in lowest terms. Units must be the same: convert as needed. DO NOT change to decimal: ratios are fractions. February 9, 2017 Geometry 8.1 Similar Polygons 7
8 Simplifying Ratios 20 in 20 in 4 5 in 5 4 ft 48 in 4 12 in 12 Same Units! February 9, 2017 Geometry 8.1 Similar Polygons 8
9 Try it. Simplify: 5 hrs 300 min min 40 min 2 February 9, 2017 Geometry 8.1 Similar Polygons 9
10 Example 1 Two adjacent sides of a rectangle are in the ratio 5:3. The perimeter of the rectangle is 48 cm. Find the length and the width. Is the ratio of sides 5:3? 3 5 Yes Is the perimeter 48? No, it s 16. February 9, 2017 Geometry 8.1 Similar Polygons 10
11 Example 1 Two adjacent sides of a rectangle are in the ratio 5:3. The perimeter of the rectangle is 48 cm. Find the length and the width. Is the ratio of sides 5:3? 3x 5x Yes 5x 5 3x 3 February 9, 2017 Geometry 8.1 Similar Polygons 11
12 Example 1 Two adjacent sides of a rectangle are in the ratio 5:3. The perimeter of the rectangle is 48 cm. Find the length and the width. 3x 5x Use the perimeter formula: 2(3x + 5x) = 48 2(8x) = 48 16x = 48 x = 3 February 9, 2017 Geometry 8.1 Similar Polygons 12
13 Example 1 Two adjacent sides of a rectangle are in the ratio 5:3. The perimeter of the rectangle is 48 cm. Find the length and the width. 3x 5x = 5(3) = 15 5x February 9, 2017 Geometry 8.1 Similar Polygons 13
14 Example 1 Two adjacent sides of a rectangle are in the ratio 5:3. The perimeter of the rectangle is 48 cm. Find the length and the width. 3x 5x = 5(3) = 15 cm 3x = 3(3) = 9 cm 15 February 9, 2017 Geometry 8.1 Similar Polygons 14
15 Example 1 Two adjacent sides of a rectangle are in the ratio 5:3. The perimeter of the rectangle is 48 cm. Find the length and the width x = 5(3) = 15 cm 3x = 3(3) = 9 cm Perimeter: 2(15 + 9) = 2(24) = 48 February 9, 2017 Geometry 8.1 Similar Polygons 15
16 Example 1 Two adjacent sides of a rectangle are in the ratio 5:3. The perimeter of the rectangle is 48 cm. Find the length and the width. The ratio of the sides is February 9, 2017 Geometry 8.1 Similar Polygons 16
17 Extended Ratio You can compare more than two numbers in a ratio. Don t write them as fractions! The ratio of a to b to c is a:b:c. February 9, 2017 Geometry 8.1 Similar Polygons 17
18 Your Turn 1 The angles of a triangle are in the ratio 2:3:5. Find the measure of each angle. Solution: 5x 3x 2x February 9, 2017 Geometry 8.1 Similar Polygons 18
19 Your Turn 1 - Solution 2x + 3x + 5x = x = 180 x = 18 5x x 2x 36 February 9, 2017 Geometry 8.1 Similar Polygons 19
20 Proportion An equation which states that two or more ratios are equal. a b c d February 9, 2017 Geometry 8.1 Similar Polygons 20
21 Alternate Notation a b c d may also be written a:b = c:d means extremes February 9, 2017 Geometry 8.1 Similar Polygons 21
22 Cross Product Property If Means a b c d Extremes then ad bc In a proportion, the product of the means equals the product of the extremes. February 9, 2017 Geometry 8.1 Problem Solving in Geometry with Proportions 22
23 Reciprocal Property If two ratios are equal, then their reciprocals are equal. If a c b d then b a d c February 9, 2017 Geometry 8.1 Ratio and Proportion 23
24 Exchange Property If a c b d ad bc ad dc bc dc then a c b d February 9, 2017 Geometry 8.1 Problem Solving in Geometry with Proportions 24
25 Exchange Property a c a b If, then. b d c d February 9, 2017 Geometry 8.1 Problem Solving in Geometry with Proportions 25
26 Addition Property a c a b c d If, then. b d b d February 9, 2017 Geometry 8.1 Problem Solving in Geometry with Proportions 26
27 Example 2 Solve: x x x x 8 x 7.5 February 9, 2017 Geometry 8.1 Ratio and Proportion 27
28 Example 3 Solve: 4 x 5 8 5x 4 8 5x x 5 x 6.4 February 9, 2017 Geometry 8.1 Ratio and Proportion 28
29 Example 4 Solve: 3 x 1 5 2x 1 3(2x 1) 5(x 1) 6x 3 5x 5 x 8 Check: (8) February 9, 2017 Geometry 8.1 Ratio and Proportion 29
30 Your Turn x 1 2x 3 3(2x 3) 5(4x 1) 6x 9 20x x x 1 February 9, 2017 Geometry 8.1 Ratio and Proportion 30
31 Example 5 x 3 = 27 x x 2 = 81 x 2 = ± 81 x = ±9 February 9, 2017 Geometry 8.1 Ratio and Proportion 31
32 Example 6 x 2 8 = 2 x 2 (x 2) 2 = 16 (x 2) 2 = ± 16 x 2 = ±4 x 2 = 4 x = 6 or x 2 = 4 x = 2 February 9, 2017 Geometry 8.1 Ratio and Proportion 32
33 Example 7 x 3 = x 15 x 7 x 2 7x = 3x + 45 x 2 4x 45 = 0 x 9 (x + 5) = 0 x 9 = 0 x + 5 = 0 x = 9 or x = 5 February 9, 2017 Geometry 8.1 Ratio and Proportion 33
34 Similar Polygons Two polygons are similar if and only if: Corresponding Angles are congruent. Corresponding Sides are proportional. Use the symbol ~ for similar. To show that two polygons are similar, you must prove both things: angles congruent, sides proportional. February 9, 2017 Geometry 8.1 Similar Polygons 34
35 For example ABCD and RSTV are similar polygons. This means: Corresponding angles are congruent. Corresponding sides are proportional. B 9 6 S A R T 15 C 8 12 V D February 9, 2017 Geometry 8.1 Similar Polygons 35
36 Similar Polygons The similarity statement is: ABCD ~ RSTV A 9 B 9 R 6 S C 10 V 8 T D February 9, 2017 Geometry 8.1 Similar Polygons 36
37 Similar Polygons Corresponding angles are congruent: A R, B S, C T, D V A 9 B 9 R 6 S C 10 V 8 T D February 9, 2017 Geometry 8.1 Similar Polygons 37
38 Similar Polygons Corresponding sides are proportional: A 15 9 D B 12 9 C February 9, 2017 Geometry 8.1 Similar Polygons 38 R 10 6 V S 8 6 T
39 Example 8 List the congruent angles. Write the ratios of the corresponding sides. K J 70 J Q, K S, L R JK JL KL QS QR SR S 70 L Q R JKL ~ QSR February 9, 2017 Geometry 8.1 Similar Polygons 39
40 Example 9 Are these figures similar? Yes Why? Corr. angles congruent N H 2 E F 1.5 G M Corr. sides proportional. 4 3 HE ON = 2 4 = 1 2 ; EF NM = 3 6 = 1 2 ; FG MP = = = 1 2 ; GH PO = 4 8 = 1 2 O 8 P February 9, 2017 Geometry 8.1 Similar Polygons 40
41 Example 9 Write the similarity statements. E N, F M G P, H O EF FG GH HE = NM MP PO ON 4 N H 2 E F 1.5 G M 3 EFGH ~ NMPO O 8 P Or: HEFG ~ ONMP, GFEH ~ PMNO, EHGF ~ NOPM, etc. February 9, 2017 Geometry 8.1 Similar Polygons 41
42 Your Turn 9 You want to print a picture from your camera. You have two sizes of paper for your printer: 4 6 and 5 7. Does it matter? Will the pictures printed from each size of paper be similar? Sides not proportional, figures not similar. February 9, 2017 Geometry 8.1 Similar Polygons 42
43 Similarity Ratio The term similarity ratio describes the ratio of corresponding sides of similar polygons. It is also known as the ratio of similarity. The similarity ratio is often called the scale factor. February 9, 2017 Geometry 8.1 Similar Polygons 43
44 Ratio of Similarity The similarity ratio of JKL to QSR is 10/5 or 2/1. The similarity ratio of QSR to JKL is 5/10 or 1/2. K 10 J 70 S 5 70 L Q R February 9, 2017 Geometry 8.1 Similar Polygons 44
45 Scale Factor The scale factor of JKL to QSR is 10/5 or 2/1. The scale factor of QSR to JKL is 5/10 or 1/2. K 10 J 70 S 5 70 L Q R February 9, 2017 Geometry 8.1 Similar Polygons 45
46 Corresponding Lengths in Similar Polygons If two polygons are similar, then the ratio of any two corresponding lengths in the polygons is equal to the scale factor of the similar polygons. February 9, 2017 Geometry 8.1 Similar Polygons 46
47 This means If any two polygons are similar, not only do the sides have the same scale factor, then so do the: Altitudes Medians Diagonals And any corresponding lengths. February 9, 2017 Geometry 8.1 Similar Triangles 47
48 Example 10 MAD ~ CAP M D Find x P A 10 x C February 9, 2017 Geometry 8.1 Similar Triangles 48
49 Example 10 Solution Since MAD ~ CAP, sides and altitudes are proportional: sides altitudes M D 10 x A 24x 200 x P x 10 C February 9, 2017 Geometry 8.1 Similar Triangles 49
50 Your Turn 10 The figures are similar. Find the length of the diagonal of the larger one d February 9, 2017 Geometry 8.1 Similar Triangles 50
51 Your Turn 10 Solution sides d 3d 64 d 8 21 diagonals 1 3 ~ 8 d February 9, 2017 Geometry 8.1 Similar Triangles 51
52 Example 11 Solve for x and y if the triangles are similar. x y 2 6 February 9, 2017 Geometry 8.1 Similar Polygons 52
53 Example 11 Solution x + 6 Solve for x 20 y 2 20 x ( x 6) 80 x 6 10 x 4 Scale Factor is 20/ y ( y 2) 120 y 2 15 y 17 Solve for y February 9, 2017 Geometry 8.1 Similar Polygons 53
54 Your Turn 11 Find x and y if the figures are similar x y February 9, 2017 Geometry 8.1 Similar Polygons 54
55 Your Turn 11 Solution x Similarity Ratio y = y = y 60 x x x x 70 February 9, 2017 Geometry 8.1 Similar Polygons 55
56 Example 12 ABC ~ RST. AB = 20, ST = 4, BC = RS. Find BC and RS. A 20 R x B x C S 4 T February 9, 2017 Geometry 8.1 Similar Polygons 56
57 Example 12 Solution AB RS 20 x 2 x x x x BC ST x B A x C x S R 4 T ABC ~ RST Remember algebra? Why didn t we use ± 80? This is Geometry and lengths can t be negative. February 9, 2017 Geometry 8.1 Similar Polygons 57
58 Perimeter and Similar Figures Given ABCD ~ FGHI 1. Find the scale factor of ABCD to FGHI. The only known corresponding sides are AB and FG. 9 6 = 3 2 A F x z y B 9 C G 6 H February 9, 2017 Geometry 8.1 Similar Polygons 58
59 Perimeter and Similar Figures 2. Find the values of x, y, and z. 3 2 = 15 x 3x = 30 x = B A = 18 y 3y = 36 y = C 10x z y 12 8 G F = 12 z 3z = 24 z = 8 H February 9, 2017 Geometry 8.1 Similar Polygons 59
60 Perimeter and Similar Figures 3. Find the perimeters of ABCD and FGHI. P = 42 P = 28 A F B 9 C G 6 H February 9, 2017 Geometry 8.1 Similar Polygons 60
61 Perimeter and Similar Figures 4. Find the ratio of the perimeters. Ratio of perimeters = = 3 2 Ratio of Similarity P = 42 P = B A C G F H February 9, 2017 Geometry 8.1 Similar Polygons 61
62 Perimeter and Similar Figures 5. Find the areas of ABCD and FGHI. A = 1 2 bh A ABC = A ABC = 6 9 A ABC = 54 A F A FGH = A FGH = 3 8 A FGH = B 9 C G 6 H February 9, 2017 Geometry 8.1 Similar Polygons 62
63 Perimeter and Similar Figures 6. Find the ratio of the areas. A ABC = 54 A FGH = 24 = 9 4 = Ratio of Similarity A B 9 C 3 2 F G 6 H February 9, 2017 Geometry 8.1 Similar Polygons 63
64 Theorem 8.1 If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths. Similarity Ratio = a b Perimeter Ratio = a b February 9, 2017 Geometry 8.1 Similar Polygons 64
65 Theorem 8.2 If two polygons are similar, then the ratio of their areas is equal to the square of the ratios of their corresponding side lengths. Similarity Ratio = a b Area Ratio = a2 b 2 February 9, 2017 Geometry 8.1 Similar Polygons 65
66 Example These figures are similar. Find the perimeter and area of the smaller one. Similarity ratio = 20 8 = 5 2 P = 100 P = 40? A = = 100 P 5P = 200 P = 40 February 9, 2017 Geometry 8.1 Similar Polygons 66
67 Example Similarity ratio = 5 2 Area ratio = 52 = P = 100 P = 40 A = 375 A = 60? = 375 A 25A = 1500 P = 60 February 9, 2017 Geometry 8.1 Similar Polygons 67
68 Your Turn 14 Given MNOP ~ QRST, with MN = 8 and QR = 12. MNOP has a perimeter of 24 and area of 56. Find the perimeter and area of QRST. MN QR = 8 12 = = 24 P 2P = 72 P = 36 Area Ratio = 22 = = 56 A 4A = 504 A = 126 February 9, 2017 Geometry 8.1 Similar Polygons 68
69 Your Turn 14 In the diagram, GHJK LMNP. Find the perimeter and area of LMNP. Perimeter of GHJK = 38 m Area of GHJK = 84 m 2 s. f. = 7 21 = = 38 P P= 3 38 A= 114 m 2 February 9, 2017 Geometry 8.1 Similar Polygons 69
70 Your Turn 14 In the diagram, GHJK LMNP. Find the perimeter and area of LMNP. Perimeter of GHJK = 38 m Area of GHJK = 84 m 2 s. f. = 7 21 = 1 3 Area of GHJK = (s. f. )2 Area of LMNP 84 A = A = A= 9 84 A= 756 m 2 February 9, 2017 Geometry 8.1 Similar Polygons 70
71 Summary Two polygons are similar if they have the same shape, but a different size. If polygons are similar corresponding angles are congruent, and corresponding sides are proportional. The ratio of any two corresponding sides is the scale factor. If two similar figures have a similarity ratio of a, then the ratio of the perimeters is a. b b If two similar figures have a similarity ratio of a b, then the ratio of the areas is a 2 b 2. February 9, 2017 Geometry 8.1 Similar Polygons 71
72 Assignment February 9, 2017 Geometry 8.1 Similar Polygons 72
8.1 Day 1 Warmup. Solve each equation. 1. 4x + 5x + 6x = (x 5) 2 = 81. in simplest form. 3. Write 16
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