Warm Up 1. If ∆QRS  ∆ZYX, identify the pairs of congruent angles and the pairs of congruent sides. Solve each proportion. 2. 3. Q  Z; R  Y; S  X; QR  ZY; RS  YX; QS  ZX x = 9 x = 18

California Standards NS1.3 Use proportions to solve problems (e.g., determine the value of N if = , find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. N 21 4 7

Vocabulary similar similar polygons similarity ratio corresponding side/angles indirect measurement

Similar figures: figures that have the same shape but not necessarily the same size. Dilation: when a figure is enlarged to be similar to another figure. Reduction: when a figure is made smaller it also produces similar figures. Indirect measurement is a method of using proportions to find an unknown length or distance in similar figures.

Similar polygons are polygons in which: Definition: Similar polygons are polygons in which: The ratios of the measures of corresponding sides are equal. Corresponding angles are congruent. Figures that are similar (~) have the same shape but not necessarily the same size.

Two polygons are similar polygons if and only if their corresponding angles are congruent and their corresponding side lengths are proportional. Writing a similarity statement is like writing a congruence statement—be sure to list corresponding vertices in the same order. Writing Math

Ratio A comparison of two quantities using division 3 ways to write a ratio: a to b a : b

Proportion An equation stating that two ratios are equal Example: Cross products: means and extremes a and d = extremes b and c = means ad = bc When you work with proportions, be sure the ratios compare corresponding measures. Helpful Hint

3/5 3 to 5 3 : 5 A comparison of two quantities: the ratio of 3 to 5 means ‘3 for every 5.’ 3/5  3 to 5 3 : 5 Ratios whose fraction representations are equivalent are called equivalent ratios. An equation stating that two ratios are equal.

Proportions and Similar Figures ∆ABC ~ ∆DFE: Find the length of DE. Define x as DE: Substitute in known values. Solve proportion.

Apply Similarity (Real-world) A tree casts a shadow 7.5 feet long. A woman 5 ft tall casts a shadow 3 ft long. The triangle show for the tree and its shadow is similar to the triangle shown for the woman and her shadow. How tall is the tree? Label the diagram with known information and a variable to show the unknown Write a proportion statement and solve. x 5

Example 3: Hobby Application Find the length of the model to the nearest tenth of a centimeter. Let x be the length of the model in centimeters. The rectangular model of the racing car is similar to the rectangular racing car, so the corresponding lengths are proportional.

Example 3 Continued 5(6.3) = x(1.8) Cross Products Prop. 31.5 = 1.8x Simplify. 17.5 = x Divide both sides by 1.8. The length of the model is 17.5 centimeters.

Check It Out! Example 3 A boxcar has the dimensions shown. A model of the boxcar is 1.25 in. wide. Find the length of the model to the nearest inch.

Check It Out! Example 3 Continued Cross Products Prop. 45.3 = 9x Simplify. 5  x Divide both sides by 9. The length of the model is approximately 5 inches.

Additional Example 1: Finding Unknown Lengths in Similar Figures Find the unknown length in the similar figures. AC QS AB QR = Write a proportion using corresponding sides. 12 48 14 w = Substitute lengths of the sides. 12 · w = 48 · 14 Find the cross product. 12w = 672 Multiply. 12w 12 672 12 = Divide each side by 12. w = 56 QR is 56 centimeters.

If a line is parallel to one side of a triangle and intersects the other two sides of the triangle, then it separates those sides into proportional parts. A B C X Y *If XY ll CB, then

Find the unknown length in the similar figures. Check It Out! Example 1 Find the unknown length in the similar figures. A B C D 10 cm 12 cm Q R T 24 cm x S 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. x = 20 QR is 20 centimeters.

Additional Example 2: 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 product. 8x = 24 Multiply. 8x 8 24 8 = Divide each side by 8. x = 3 The base of the inside triangle is 3 inches.

Check It Out! Example 2 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 product. 6w = 36 Multiply. 6w 6 = 36 6 Divide each side by 6. w = 6 The right rectangle is 6 cm wide.

Additional Example 3: 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 Write a proportion. 25 15 ≈ 50 h Use compatible numbers to estimate. 53 ≈ 50 h Simplify. 5h ≈ 150 Cross multiply. h ≈ 30 Divide each side by 5. The traffic light is about 30 feet high.

Check It Out! Example 3 The inside triangle is similar in shape to the outside triangle. Find the height of the outside triangle. 5 14.75 = h 30.25 Write a proportion. h ft 5 15 h 30 Use compatible numbers to estimate. ≈ 5 ft 13 h 30 14.75 ft ≈ Simplify. 30.25 ft 1 • 30 ≈ 3 • h Cross multiply. 30 ≈ 3h Multiply. 10 ≈ h Divide each side by 3. The outside triangle is about 10 feet tall.