Presentation – Six Lessons 5-7 Similar Figures and Proportions Course 2 Warm Up Problem of the Day Presentation – Six Lessons

Similar Figures and Proportions Course 2 5-7 Similar Figures and Proportions Warm Up Find the cross products, then tell whether the ratios are equal. 16 6 , 40 15 1. 240 = 240; equal 3 8 18 46 , 2. 138 = 144; not equal 8 9 , 24 27 3. 216 = 216; equal 28 12 , 42 18 4. 504 = 504; equal

Similar Figures and Proportions Course 2 5-7 Similar Figures and Proportions Problem of the Day Every 8th telephone pole along a road has a red band painted on it. Every 14th pole has an emergency call phone on it. What is the number of the first pole with both a red band and a call phone? 56

Similar Figures and Proportions Course 2 5-7 Similar Figures and Proportions Lesson 1 EQ: How can I determine if two figures are similar?

Insert Lesson Title Here Course 2 5-7 Similar Figures and Proportions Insert Lesson Title Here Vocabulary Words similar corresponding sides corresponding angles

Similarity in the Real World Course 2 5-7 Similar Figures and Proportions Similarity in the Real World Octahedral fluorite is a crystal found in nature. It grows in the shape of an octahedron, which is a solid figure with eight triangular faces. The triangles in different-sized fluorite crystals are similar figures. Similar figures have the same shape but not necessarily the same size.

Vocabulary 5-7 Similar Figures and Proportions SIMILAR FIGURES Course 2 5-7 Similar Figures and Proportions Vocabulary SIMILAR FIGURES Two figures are similar if The measures of their corresponding angles are equal. The ratios of the lengths of the corresponding sides are proportional.

Vocabulary 5-7 Similar Figures and Proportions Course 2 5-7 Similar Figures and Proportions Vocabulary Matching sides of two or more polygons are called corresponding sides, and matching angles are called corresponding angles. 82◦ Corresponding angles D E F Corresponding sides A B C 55◦ 43◦

Symbols 5-7 Math Dictionary: SYMBOLS ∆ABC AB ║ and ~ Math Tip: Course 2 5-7 Math Dictionary: SYMBOLS When naming similar figures, list the letters of the corresponding vertices in the same order. In the previous table ∆ABC ~ ∆DEF. Math Tip: Symbols ∆ABC AB ║ and ~

Similar Figures and Proportions Course 2 5-7 Similar Figures and Proportions A side of a figure can be named by its endpoints, with a bar above. AB Without the bar, the letters indicate the length of the side. Reading Math

Example 1: Determining Whether Two Triangles Are Similar Course 2 5-7 Similar Figures and Proportions Example 1: Determining Whether Two Triangles Are Similar Identify the corresponding sides in the pair of triangles. Then use ratios to determine whether the triangles are similar. E AB corresponds to DE. 16 in 10 in A C 28 in BC corresponds to EF. 4 in D 7 in 40 in F AC corresponds to DF. B AB DE = ? BC EF = ? AC DF Step 1: Write ratios using the corresponding sides. 4 16 = ? 7 28 = ? 10 40 Step 2: Substitute the length of the sides. 1 4 = ? 1 4 = ? 1 4 Step 3: Simplify each ratio. Since the ratios of the corresponding sides are equivalent, the triangles are similar.

Similar Figures and Proportions Course 2 5-7 Similar Figures and Proportions Check It Out: Example 2 Identify the corresponding sides in the pair of triangles. Then use ratios to determine whether the triangles are similar. E AB corresponds to DE. 9 in 9 in A C 21 in BC corresponds to EF. 3 in D 7 in 27 in F AC corresponds to DF. B AB DE = ? BC EF = ? AC DF Write ratios using the corresponding sides. 3 9 = ? 7 21 = ? 9 27 Substitute the length of the sides. 1 3 = ? 1 3 = ? 1 3 Simplify each ratio. Since the ratios of the corresponding sides are equivalent, the triangles are similar.

Similar Figures and Proportions Course 2 5-7 Similar Figures and Proportions Lesson 2 EQ: How can I determine if figures are similar based on their angle measure?

How can I determine if these shapes are similar? Course 2 5-7 Similar Figures and Proportions How can I determine if these shapes are similar? Tell whether the figures are similar. Yes….The corresponding angles of the figures have equal measure. D 60° F E A 60° *remember – the sum of the interior angles of a triangle = 180° C B

Similar Figures and Proportions Insert Lesson Title Here Course 2 5-7 Similar Figures and Proportions Insert Lesson Title Here Try One: Tell whether the figures are similar. (Notice the shapes are turned) 1. 59° 35° 86° similar

Similar Figures and Proportions Insert Lesson Title Here Course 2 5-7 Similar Figures and Proportions Insert Lesson Title Here Try another: Tell whether the figures are similar. 2. 119° 55° 107° 79° 80° 135° 38° not similar

Similar Figures and Proportions Course 2 5-7 Similar Figures and Proportions Lesson 3 EQ: How can I determine the scale factor of similar figures?

Vocabulary 5-7 Similar Figures and Proportions Course 2 5-7 Similar Figures and Proportions Vocabulary Scale Factor: The ratio of the lengths of corresponding sides in similar figures

How can I determine the scale factor of similar figures? Course 2 5-7 Similar Figures and Proportions How can I determine the scale factor of similar figures? EXAMPLE 1 The figures below are similar 9 12 3 4

How can I determine the scale factor of similar figures? Course 2 5-7 Similar Figures and Proportions How can I determine the scale factor of similar figures? EXAMPLE 2 A ~ B 2.5 A 5 B 2 1

Similar Figures and Proportions Course 2 5-7 Similar Figures and Proportions Lesson 4 EQ: What is the relationship between the scale factor, side lengths, perimeter, and area?

*The scale factor SQUARED tells you the ratio of the areas The relationship between scale factor, side lengths, perimeter, and area... Figure A B Ratio/Scale Factor Corresponding Sides Side Lengths (feet) Perimeter Area 5.5 ft 11 ft 3 ft 6 ft 5 ft 10 ft *The scale factor tells you the ratio of corresponding side lengths and the ratio of the perimeters *The scale factor SQUARED tells you the ratio of the areas

Similar Figures and Proportions Course 2 5-7 Similar Figures and Proportions Lesson 5 EQ: How can I determine missing side lengths of similar figures?

Example 1: Missing Side Lengths Course 2 5-8 Using Similar Figures Example 1: Missing Side Lengths Find the unknown length in similar figures. AC QS = AB QR Step 1: Write a proportion using corresponding sides. 12 48 14 w = Step 2: Substitute lengths of the sides. 12 · w = 48 · 14 Step 3: Cross multiply and divide. 12w = 672 12w 12 672 12 = Divide each side by 12 to isolate the variable. w = 56 QR is 56 centimeters.

Insert Lesson Title Here Course 2 5-8 Using Similar Figures Insert Lesson Title Here Check It Out: Example 2 Find the unknown length in similar figures. x 10 cm Q R A B 12 cm 24 cm D C S T AC QS AB QR = Write a proportion using corresponding sides. 12 24 10 x = Substitute lengths of the sides. 12 · x = 24 · 10 Find the cross product. 12x = 240 Multiply. 12x 12 240 12 = Divide each side by 12 to isolate the variable. x = 20 QR is 20 centimeters.

Insert Lesson Title Here Course 2 5-8 Using Similar Figures Insert Lesson Title Here Example 3: Measurement Application The inside triangle is similar in shape to the outside triangle. Find the length of the base of the inside triangle. Let x = the base of the inside triangle. 8 2 12 x Write a proportion using corresponding side lengths. = 8 · x = 2 · 12 Find the cross products. 8x = 24 Multiply. 8x 8 24 8 = Divide each side by 8 to isolate the variable. x = 3 The base of the inside triangle is 3 inches.

Insert Lesson Title Here Course 2 5-8 Using Similar Figures Insert Lesson Title Here Example 4 The rectangle on the left is similar in shape to the rectangle on the right. Find the width of the right rectangle. 12 cm 6 cm 3 cm ? Let w = the width of the right rectangle. 6 12 3 w Write a proportion using corresponding side lengths. = 6 ·w = 12 · 3 Find the cross products. 6w = 36 Multiply. 6w 6 = 36 6 Divide each side by 6 to isolate the variable. w = 6 The right rectangle is 6 cm wide.

Insert Lesson Title Here Course 2 5-8 Using Similar Figures Insert Lesson Title Here Ticket-out-the-door Find the unknown length in each pair of similar figures. 1. 2. 28

Insert Lesson Title Here Course 2 5-8 Using Similar Figures Insert Lesson Title Here Ticket-out-the-door Find the unknown length in each pair of similar figures. 3. The width of the smaller rectangular cake is 5.75 in. The width of a larger rectangular cake is 9.25 in. Estimate the length of the larger rectangular cake. 29

Similar Figures and Proportions Course 2 5-7 Similar Figures and Proportions Lesson 6 EQ: How can I use shadow math to find missing side lengths?

Similar Figures and Proportions Course 2 5-7 Similar Figures and Proportions Example 1: Missing Side Lengths Step 1: Label Corresponding Parts. Step 2: Write a Proportion. Step 3: Cross multiply and divide. x 1.5m 5m 1m

Additional Example 2: Estimating with Indirect Measurement Course 2 5-8 Using Similar Figures Additional Example 2: Estimating with Indirect Measurement City officials want to know the height of a traffic light. Estimate the height of the traffic light. 27.25 15 48.75 h = Step 1: Label Corresponding Parts. 27 15 49 h h ft ≈ 9 5 49 h ≈ Step 2: Write a Proportion 27.25 ft 9h ≈ 245 Step 3: Cross multiply. 48.75 ft h ≈ 27 Multiply each side by 9 to isolate the variable. The traffic light is about 30 feet tall.

5-8 Using Similar Figures Check It Out: Example 3 Course 2 5-8 Using Similar Figures Check It Out: Example 3 The inside triangle is similar in shape to the outside triangle. These are called NESTED triangles. Find the height of the outside triangle. 5 14.75 h 30.25 = Write a proportion. 5 15 h 30 Use compatible numbers to estimate. ≈ h ft 5 ft 13 h 30 ≈ Simplify. 1 • 30 ≈ 3 • h Cross multiply. 14.75 ft 30 ≈ 3h Multiply each side by 5 to isolate the variable. 30.25 ft 10 ≈ h The outside triangle is about 10 feet tall.

Classwork Problem 5.1(pg.78-79) Problem 5.2 (pg. 80-81)