Geometry 8.1 Similar Polygons

Geometry 8.1 Similar Polygons
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. December 7, 2018 Geometry 8.1 Similar Polygons

8.1 Day 2 Warmup Solve each proportion. 1. 𝑥 4 = 𝑥 9 = 4 𝑥 3. 4 − 𝑥 12 = 3 − 𝑥−3 = 8 3𝑥− 𝑥+1 = 𝑥−3 5 December 7, 2018 Geometry 8.1 Similar Polygons

8.1 Essential Question How are similar polygons related? December 7, 2018 Geometry 8.1 Similar Polygons

Goals Solve proportions. Identify similar polygons Find the ratio of similarity between similar figures. Solve problems involving similar figures. December 7, 2018 Geometry 8.1 Similar Polygons

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 December 7, 2018 Geometry 8.1 Similar Polygons

Simplifying Ratios Ratios must be in lowest terms. Units must be the same: convert as needed. DO NOT change to decimal: ratios are fractions. December 7, 2018 Geometry 8.1 Similar Polygons

Simplifying Ratios Same Units! December 7, 2018 Geometry 8.1 Similar Polygons

Try it. Simplify: December 7, 2018 Geometry 8.1 Similar Polygons

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? Yes Is the perimeter 48? No, it’s 16. 3 5 December 7, 2018 Geometry 8.1 Similar Polygons

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? Yes 3x 5x December 7, 2018 Geometry 8.1 Similar Polygons

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. Use the perimeter formula: 2(3x + 5x) = 48 2(8x) = 48 16x = 48 x = 3 3x 5x December 7, 2018 Geometry 8.1 Similar Polygons

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. 5x = 5(3) = 15 3x 5x December 7, 2018 Geometry 8.1 Similar Polygons

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. 5x = 5(3) = 15 cm 3x = 3(3) = 9 cm 3x 15 December 7, 2018 Geometry 8.1 Similar Polygons

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. 5x = 5(3) = 15 cm 3x = 3(3) = 9 cm Perimeter: 2(15 + 9) = 2(24) = 48 9 15 December 7, 2018 Geometry 8.1 Similar Polygons

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 9 15 December 7, 2018 Geometry 8.1 Similar Polygons

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. December 7, 2018 Geometry 8.1 Similar Polygons

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 December 7, 2018 Geometry 8.1 Similar Polygons

Your Turn 1 - Solution 2x + 3x + 5x = 180 10x = 180 x = 18 90 5x 54 36 3x 2x December 7, 2018 Geometry 8.1 Similar Polygons

Proportion An equation which states that two or more ratios are equal. December 7, 2018 Geometry 8.1 Similar Polygons

Alternate Notation may also be written a:b = c:d means extremes December 7, 2018 Geometry 8.1 Similar Polygons

Cross Product Property
If Means Extremes then In a proportion, the product of the means equals the product of the extremes. December 7, 2018 Geometry 8.1 Problem Solving in Geometry with Proportions

If then Reciprocal Property
If two ratios are equal, then their reciprocals are equal. If then December 7, 2018 Geometry 8.1 Ratio and Proportion

Geometry 8.1 Problem Solving in Geometry with Proportions
Exchange Property If then December 7, 2018 Geometry 8.1 Problem Solving in Geometry with Proportions

Exchange Property December 7, 2018 Geometry 8.1 Problem Solving in Geometry with Proportions

Addition Property December 7, 2018 Geometry 8.1 Problem Solving in Geometry with Proportions

Geometry 8.1 Ratio and Proportion
Example 2 Solve: December 7, 2018 Geometry 8.1 Ratio and Proportion

Example 3 Solve: December 7, 2018 Geometry 8.1 Ratio and Proportion

Example 4 Solve: Check: December 7, 2018 Geometry 8.1 Ratio and Proportion

Your Turn 4 December 7, 2018 Geometry 8.1 Ratio and Proportion

Example 5 𝑥 3 = 27 𝑥 𝑥 2 =81 𝑥 2 =± 81 𝑥=±9 December 7, 2018 Geometry 8.1 Ratio and Proportion

Example 6 𝑥−2 8 = 2 𝑥−2 (𝑥−2) 2 =16 (𝑥−2) 2 =± 16 𝑥−2=±4 𝑥−2=4 𝑥−2=−4 or 𝑥=6 𝑥=−2 December 7, 2018 Geometry 8.1 Ratio and Proportion

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 or 𝑥=9 𝑥=−5 December 7, 2018 Geometry 8.1 Ratio and Proportion

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. December 7, 2018 Geometry 8.1 Similar Polygons

For example ABCD and RSTV are similar polygons. This means: Corresponding angles are congruent. Corresponding sides are proportional. A B C D 15 12 9 R S T V 10 8 6 6 December 7, 2018 Geometry 8.1 Similar Polygons

The similarity statement is: ABCD ~ RSTV A B C D 15 12 9 R S T V 10 8 6 6 December 7, 2018 Geometry 8.1 Similar Polygons

Corresponding angles are congruent: A  R, B  S, C  T, D  V A B C D 15 12 9 R S T V 10 8 6 6 December 7, 2018 Geometry 8.1 Similar Polygons

Corresponding sides are proportional: A B C D 15 12 9 S 6 R 6 T 10 8 V December 7, 2018 Geometry 8.1 Similar Polygons

Example 8 List the congruent angles. Write the ratios of the corresponding sides. J  Q, K  S, L  R 70 J K L Q R S JKL ~ QSR December 7, 2018 Geometry 8.1 Similar Polygons

Example 9 Are these figures similar? Yes Why? Corr. angles congruent Corr. sides proportional. 𝐻𝐸 𝑂𝑁 = 2 4 = 1 2 ; 𝐸𝐹 𝑁𝑀 = 3 6 = 1 2 ; 𝐹𝐺 𝑀𝑃 = = = 1 2 ; 𝐺𝐻 𝑃𝑂 = 4 8 = 1 2 3 E F 2 1.5 H G 4 N 6 M 4 3 O 8 P December 7, 2018 Geometry 8.1 Similar Polygons

Example 9 Write the similarity statements.
3 E F 2 1.5 H G 4 N 6 M 4 3 O EFGH ~ NMPO 8 P Or: HEFG ~ ONMP, GFEH ~ PMNO, EHGF ~ NOPM, etc. December 7, 2018 Geometry 8.1 Similar Polygons

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? 4 × 6 Sides not proportional, figures not similar. 5 × 7 December 7, 2018 Geometry 8.1 Similar Polygons

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. December 7, 2018 Geometry 8.1 Similar Polygons

Ratio of Similarity Or... 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. 70 J K L Q R S 10 5 December 7, 2018 Geometry 8.1 Similar Polygons

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. 70 J K L Q R S 10 5 December 7, 2018 Geometry 8.1 Similar Polygons

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. December 7, 2018 Geometry 8.1 Similar Polygons

Geometry 8.1 Similar Triangles
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. December 7, 2018 Geometry 8.1 Similar Triangles

Example 10 MAD ~ CAP Find x. M A D C P 24 10 20 x December 7, 2018 Geometry 8.1 Similar Triangles

Example 10 Solution Since MAD ~ CAP, sides and altitudes are proportional: M A D C P 24 10 20 x sides altitudes December 7, 2018 Geometry 8.1 Similar Triangles

Your Turn 10 The figures are similar. Find the length of the diagonal of the larger one. d 8 8 3 December 7, 2018 Geometry 8.1 Similar Triangles

Your Turn 10 Solution sides diagonals d ~ 8 8 3 December 7, 2018 Geometry 8.1 Similar Triangles

Example 11 Solve for x and y if the triangles are similar. 20 x + 6 8 4 y – 2 6 December 7, 2018 Geometry 8.1 Similar Polygons

Example 11 Solution Scale Factor is 20/8 20 x + 6 8 4 y – 2 6 Solve for x Solve for y December 7, 2018 Geometry 8.1 Similar Polygons

Your Turn 11 Find x and y if the figures are similar. x + 10 85 100 32 60 24 y 95 December 7, 2018 Geometry 8.1 Similar Polygons

Your Turn 11 Solution 60 x + 10 95 85 100 y 24 32 Similarity Ratio y = 360 - 100 - 85 - 95 y = 80 December 7, 2018 Geometry 8.1 Similar Polygons

Example 12 ABC ~ RST. AB = 20, ST = 4, BC = RS. Find BC and RS. A B C R S T 20 x 4 x December 7, 2018 Geometry 8.1 Similar Polygons

Example 12 Solution A B C R S T 20 4 x ABC ~ RST Remember algebra? Why didn’t we use ± 80 ? This is Geometry and lengths can’t be negative. December 7, 2018 Geometry 8.1 Similar Polygons

Perimeter and Similar Figures
Given ABCD ~ FGHI Find the scale factor of ABCD to FGHI. The only known corresponding sides are AB and FG. 9 6 = 3 2 G 12 9 18 15 6 z y x A B C F H December 7, 2018 Geometry 8.1 Similar Polygons

2. Find the values of x, y, and z. 3 2 = 18 𝑦 3𝑦=36 𝑦=12 3 2 = 15 𝑥 3𝑥=30 𝑥=10 3 2 = 12 𝑧 3𝑧=24 𝑧=8 G 12 9 18 15 6 z y x A B C F H 12 10 8 December 7, 2018 Geometry 8.1 Similar Polygons

3. Find the perimeters of ABCD and FGHI. P = 42 P = 28 G 12 9 18 15 6 8 10 A B C F H December 7, 2018 Geometry 8.1 Similar Polygons

4. Find the ratio of the perimeters. Ratio of perimeters = 2∙3∙7 2∙2∙7 = 3 2 Ratio of Similarity P = 42 P = 28 G 12 9 18 15 6 8 10 A B C F H December 7, 2018 Geometry 8.1 Similar Polygons

5. Find the areas of ABCD and FGHI. 𝐴= 1 2 𝑏ℎ 𝐴 ∆𝐴𝐵𝐶 = 1 2 9∙12 𝐴 ∆𝐴𝐵𝐶 =6∙9 𝐴 ∆𝐴𝐵𝐶 =54 𝐴 ∆𝐹𝐺𝐻 = 1 2 6∙8 𝐴 ∆𝐹𝐺𝐻 =3∙8 𝐴 ∆𝐹𝐺𝐻 =24 G 12 9 18 15 6 8 10 A B C F H December 7, 2018 Geometry 8.1 Similar Polygons

6. Find the ratio of the areas. 𝐴 ∆𝐴𝐵𝐶 =54 = 9 4 = 𝐴 ∆𝐹𝐺𝐻 =24 Ratio of Similarity G 12 9 18 15 6 8 10 A B C F H December 7, 2018 Geometry 8.1 Similar Polygons

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 December 7, 2018 Geometry 8.1 Similar Polygons

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 = 𝑎 2 𝑏 2 December 7, 2018 Geometry 8.1 Similar Polygons

Example 13 These figures are similar. Find the perimeter and area of the smaller one. Similarity ratio = = 5 2 20 8 5 2 = 100 𝑃 5𝑃=200 𝑃=40 P = 100 P = 40 P = ? A = 375 December 7, 2018 Geometry 8.1 Similar Polygons

Example 13 Similarity ratio = 5 2 Area ratio= = 25 4 20 8 25 4 = 375 𝐴 25𝐴=1500 𝑃=60 P = 100 P = 40 A = 375 A = ? A = 60 December 7, 2018 Geometry 8.1 Similar Polygons

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 = 2 3 2 3 = 24 𝑃 2𝑃=72 𝑃=36 Area Ratio = = 4 9 4 9 = 56 𝐴 4𝐴=504 𝐴=126 December 7, 2018 Geometry 8.1 Similar Polygons

Your Turn 14 In the diagram, GHJK ∼ LMNP. Find the perimeter and area of LMNP. Perimeter of GHJK = 38 m Area of GHJK = 84 m2 𝑠.𝑓.= 7 21 = 1 3 1 3 = 38 𝑃 P= 3 ∙ 38 A= 114 𝑚 2 December 7, 2018 Geometry 8.1 Similar Polygons

Your Turn 14 In the diagram, GHJK ∼ LMNP. Find the perimeter and area of LMNP. Perimeter of GHJK = 38 m Area of GHJK = 84 m2 𝑠.𝑓.= 7 21 = 1 3 𝐴𝑟𝑒𝑎 𝑜𝑓 𝐺𝐻𝐽𝐾 𝐴𝑟𝑒𝑎 𝑜𝑓 𝐿𝑀𝑁𝑃 = (𝑠.𝑓.) 2 84 𝐴 = 84 𝐴 = 1 9 A= 9 ∙ 84 A= 756 𝑚 2 December 7, 2018 Geometry 8.1 Similar Polygons

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 𝑏 , then the ratio of the perimeters is 𝑎 𝑏 . If two similar figures have a similarity ratio of 𝑎 𝑏 , then the ratio of the areas is 𝑎 𝑏 2 . December 7, 2018 Geometry 8.1 Similar Polygons

Assignment December 7, 2018 Geometry 8.1 Similar Polygons

Geometry 8.1 Similar Polygons

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