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 16, 2016 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 2x+1 9 February 16, 2016 Geometry 8.1 Similar Polygons 2
3 Geometry 8.1 Similar Polygons
4 8.1 Essential Question How are similar polygons related? February 16, 2016 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 16, 2016 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 16, 2016 Geometry 8.1 Similar Polygons 6
7 Simplifying Ratios 20 in 20 in 45 in 5 4 ft 48 in 4 12 in 12 Same Units! February 16, 2016 Geometry 8.1 Similar Polygons 7
8 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 16, 2016 Geometry 8.1 Similar Polygons 8
9 Try it. Simplify: 5 hrs 300 min min 40 min 2 February 16, 2016 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. 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 16, 2016 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. Find the length and the width. Is the ratio of sides 5:3? 3x 5x Yes 5x 5 3x 3 February 16, 2016 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. Find the length and the width. 3x 5x Use the perimeter formula: 2(3x + 5x) = 48 2(8x) = 48 16x = 48 x = 3 February 16, 2016 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. Find the length and the width. 3x Substitute the x value: 5x = 5(3) = 15 5x February 16, 2016 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. Find the length and the width. 3x Substitute the x value: 5x = 5(3) = 15 3x = 3(3) = 9 15 February 16, 2016 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. Find the length and the width Substitute the x value: 5x = 5(3) = 15 3x = 3(3) = 9 Perimeter: 2(15 + 9) = 2(24) = 48 February 16, 2016 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. Find the length and the width. The ratio of the sides is February 16, 2016 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 16, 2016 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 16, 2016 Geometry 8.1 Similar Polygons 18
19 Your Turn 1 - Solution 2x + 3x + 5x = x = 180 x = 18 5x x 2x 36 February 16, 2016 Geometry 8.1 Similar Polygons 19
20 Proportion An equation which states that two or more ratios are equal. a b c d February 16, 2016 Geometry 8.1 Similar Polygons 20
21 Alternate Notation a b c d may also be written a:b = c:d means extremes February 16, 2016 Geometry 8.1 Similar Polygons 21
22 Cross Product Property Means a b c d Extremes ad bc In a proportion, the product of the means equals the product of the extremes. February 16, 2016 Geometry 8.1 Problem Solving in Geometry with Proportions 22
23 Reciprocal Property If two ratios are equal, then their reciprocals are equal. a c b d b d a c February 16, 2016 Geometry 8.1 Ratio and Proportion 23
24 Exchange Property a c b d ad bc ad dc a c bc dc b d February 16, 2016 Geometry 8.1 Problem Solving in Geometry with Proportions 24
25 Exchange Property a c a b If, then. b d c d February 16, 2016 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 16, 2016 Geometry 8.1 Problem Solving in Geometry with Proportions 26
27 Example 2 Solve: x x x x 8 x 7.5 February 16, 2016 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 16, 2016 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 16, 2016 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 16, 2016 Geometry 8.1 Ratio and Proportion 30
31 Example 5 x 3 = 27 x x 2 = 81 x 2 = ± 81 x = ±9 February 16, 2016 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 16, 2016 Geometry 8.1 Ratio and Proportion 32
33 Similar Polygons ABCD and RSTV are similar polygons: Corresponding angles are congruent. Corresponding sides are proportional. 9 A 15 D B S R 6 10 T C 8 V February 16, 2016 Geometry 8.1 Similar Polygons 33
34 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 16, 2016 Geometry 8.1 Similar Polygons 34
35 Similar Polygons Corresponding sides are proportional: A 15 9 D B 12 9 C February 16, 2016 Geometry 8.1 Similar Polygons 35 R 10 6 V S 8 6 T
36 Similar Polygons Corr. s Sides prop. ABCD ~ RSTV A 9 B 9 R 6 S C 10 V 8 T D February 16, 2016 Geometry 8.1 Similar Polygons 36
37 Similar Polygons 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 16, 2016 Geometry 8.1 Similar Polygons 37
38 Similarity Statements List the congruent angles. Write the ratios of the corresponding sides. K J 70 S 70 L Q R J Q, K S, L R JK JL KL QS QR SR JKL QSR February 16, 2016 Geometry 8.1 Similar Polygons 38
39 Example 7 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 3 ; GH PO = 4 8 = 1 2 O 8 P February 16, 2016 Geometry 8.1 Similar Polygons 39
40 Example 7 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 16, 2016 Geometry 8.1 Similar Polygons 40
41 Your Turn 7 You want to print a picture in 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 16, 2016 Geometry 8.1 Similar Polygons 41
42 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 16, 2016 Geometry 8.1 Similar Polygons 42
43 Ratio of Similarity The ratio of similarity of JKL to QSR is 10/5 or 2/1. The ratio of similarity of QSR to JKL is 5/10 or 1/2. K 10 J 70 S 5 70 L Q R February 16, 2016 Geometry 8.1 Similar Polygons 43
44 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 16, 2016 Geometry 8.1 Similar Polygons 44
45 Generalization If any two polygons are similar, the sides are proportional. This also can be extended to: Altitudes Medians Diagonals any corresponding lengths. February 16, 2016 Geometry 8.1 Similar Triangles 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 16, 2016 Geometry 8.1 Similar Polygons 46
47 Example 8 MAD ~ CAP M D Find x P A 10 x C February 16, 2016 Geometry 8.1 Similar Triangles 47
48 Example 8 Solution Since MAD ~ CAP, sides and altitudes are proportional: sides altitudes M D 10 x A 24x 200 x P x 10 C February 16, 2016 Geometry 8.1 Similar Triangles 48
49 Your Turn 8 The figures are similar. Find the length of the diagonal of the larger one d February 16, 2016 Geometry 8.1 Similar Triangles 49
50 Your Turn 8 Solution sides d 3d 64 d 8 21 diagonals 1 3 ~ 8 d February 16, 2016 Geometry 8.1 Similar Triangles 50
51 Example 9 Solve for x and y if the triangles are similar. x y 2 6 February 16, 2016 Geometry 8.1 Similar Polygons 51
52 Example 9 Solution x y 2 Solve 20 x 6 8for x4 8( x 6) 80 x 6 10 x 4 Scale Factor is 20/ Solve 20 y 2 8 for y6 8( y 2) 120 y 2 15 y 17 February 16, 2016 Geometry 8.1 Similar Polygons 52
53 Your Turn 9 Find x and y if the figures are similar x y February 16, 2016 Geometry 8.1 Similar Polygons 53
54 Your Turn 9 Solution x Similarity Ratio y = y = y 60 x x x x 70 February 16, 2016 Geometry 8.1 Similar Polygons 54
55 Example 10 ABC ~ RST AB = 20 ST = 4 BC = RS Find BC and RS. February 16, 2016 Geometry 8.1 Similar Polygons 55
56 Example 10 Solution ABC ~ RST A AB = 20 ST = 4 20 R BC = RS x Find BC and RS. B x C S 4 T February 16, 2016 Geometry 8.1 Similar Polygons 56
57 Example 10 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 16, 2016 Geometry 8.1 Similar Polygons 57
58 Perimeter and Similar Figures 1. ABCD ~ FGHI 2. Find the scale factor of ABCD to FGHI. 3. Find the values of x, y, and z. 4. Find the perimeter of ABCD and FGHI. 5. Find the ratio of the perimeters. A 10 B F x 5 G y D 4 C I z H February 16, 2016 Geometry 8.1 Similar Polygons 58
59 Perimeter and Similar Figures 2. Find the scale factor from ABCD to FGHI. The only known corresponding 10 2 sides are AB and FG. 5 1 A 10 B F 5 G x y D 4 C I z H February 16, 2016 Geometry 8.1 Similar Polygons 59
60 Perimeter and Similar Figures 3. Find the values of x, y, and z x 1 y 1 z 10x 70 x 7 2y 14 y 7 2z 4 z 2 A 10 B F 5 G x y D 4 C I z H February 16, 2016 Geometry 8.1 Similar Polygons 60
61 Perimeter and Similar Figures 4. Find the perimeter of ABCD and FGHI. A P = B P = 21 F 5 G 7 7 D 4 C I 2 H February 16, 2016 Geometry 8.1 Similar Polygons 61
62 Perimeter and Similar Figures 5. Find the ratio of the perimeters. Ratio of perimeters 2:1 Ratio of Similarity 2:1 A P = B P = 21 F 5 G 7 7 D 4 C I 2 H February 16, 2016 Geometry 8.1 Similar Polygons 62
63 Theorem 8.1 If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths. February 16, 2016 Geometry 8.1 Similar Polygons 63
64 Example These figures are similar. Find the perimeter of the smaller one. P = 100 P =? P 20P 800 P 40 February 16, 2016 Geometry 8.1 Similar Polygons 64
65 Your Turn 11 MNOP has a perimeter of 24. Find the perimeter of QRST if MN = 8 and QR = 12. MN QR P 8P 288 P 36 February 16, 2016 Geometry 8.1 Similar Polygons 65
66 Example 12 In the diagram, ABC DEF. Find the area of DEF. s. f. = 10 5 = 2 1 Area of ABC = (s. f. )2 Area of DEF 36 A = A = A= 36 A= 9 cm 2 February 16, 2016 Geometry 8.1 Similar Polygons 66
67 Your Turn 12 In the diagram, GHJK LMNP. Find the area of LMNP. s. f. = 7 21 = 1 3 Area of GHJK = 84 m 2 Area of GHJK = (s. f. )2 Area of LMNP 84 A = A = A= 9 84 A= 756 m 2 February 16, 2016 Geometry 8.1 Similar Polygons 67
68 Summary Two polygons are similar if they have the same shape, but a different size. If polygons are similar, then corresponding angles are congruent, and corresponding sides are proportional. The ratio of any two corresponding sides is the scale factor. The ratio of the perimeters is equal to the ratio of corresponding sides. February 16, 2016 Geometry 8.1 Similar Polygons 68
69 Assignment February 16, 2016 Geometry 8.1 Similar Polygons 69
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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