Class Greeting

Chapter 7 – Lesson 2 Similar Polygons

Objective: The students will solve problems involving Similar Polygons.

Vocabulary similar scale factor similar polygons

Figures that are similar (~) have the same shape but not necessarily the same size.

Determine whether the pair of figures is similar. Justify your answer. Q Answer: The vertex angles are marked as 40º and 50º, so they are not congruent. Example 2-1a

Determine whether the pair of figures is similar. Justify your answer. Answer: Only one pair of angles are congruent, so the triangles are not similar. Example 2-1f

A scale factor is the ratio of the lengths of the corresponding sides of two similar polygons. The scale factor of ∆ABC to ∆DEF is , or . The scale factor of ∆DEF to ∆ABC is , or 2.

The two polygons are similar The two polygons are similar. Find the scale factor of polygon ABCDE to polygon RSTUV. Answer: Example 2-3e

The two polygons are similar The two polygons are similar. Find the scale factor of polygon TRAP to polygon ZOLD. Answer: Example 2-3g

Writing a similarity statement is like writing a congruence statement—be sure to list corresponding vertices in the same order. Writing Math

Two polygons are similar polygons if and only if their corresponding angles are congruent and their corresponding side lengths are proportional.

Determine whether the pair of figures is similar. Justify your answer. QR = 0.5 N  Q and P  R. By the Third Angles Theorem, M  T. Answer: The ratio of the measures of the corresponding sides are equal and the corresponding angles are congruent, so MNP RST.

Identify the pairs of congruent angles and corresponding sides. B  G and C  H. By the Third Angles Theorem, A  J. Answer: The ratio of the measures of the corresponding sides are equal and the corresponding angles are congruent, so JGH.

Determine whether the pair of figures is similar. Justify your answer. The corresponding angles are congruent. Determine whether corresponding sides are proportional. Answer: The ratio of the measures of the corresponding sides are equal and the corresponding angles are congruent, so Example 2-1c

Determine whether the pair of figures is similar. Justify your answer. The ratio of the measures of the corresponding sides are equal and the corresponding angles are congruent, Example 2-1e

rectangles ABCD and EFGH Example 2A: Identifying Similar Polygons Determine whether the polygons are similar. If so, write the scale factor and a similarity statement. rectangles ABCD and EFGH A  E, B  F, C  G, and D  H. Thus the scale factor is , and rect. ABCD ~ rect. EFGH.

Example 2B: Identifying Similar Polygons Determine whether the polygons are similar. If so, write the scale factor and a similarity statement. ∆ABCD and ∆EFGH P  R and S  W mW = mS = 62° mT = 180°– 2(62°) = 56° Since the corresponding angles of the triangles are not congruent, the triangles are not similar.

Thus the similarity ratio is , and ∆LMJ ~ ∆PNS. Determine if ∆JLM ~ ∆NPS. If so, write the similarity ratio and a similarity statement. N  M, L  P, S  J Thus the similarity ratio is , and ∆LMJ ~ ∆PNS.

When you work with proportions, be sure the ratios compare corresponding measures. Helpful Hint

Example 3: Hobby Application Find the length of the model to the nearest tenth of a centimeter. 5(6.3) = x(1.8) 31.5 = 1.8x 17.5 = x Answer: The length of the model is 17.5 centimeters.

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. 45.3 = 9x Answer: The length of the model is approximately 5 in.

Use the congruent angles to write the corresponding vertices in order. The two polygons are similar. Write a similarity statement. Then find x, y, and UV. Use the congruent angles to write the corresponding vertices in order. Example 2-3a

Now write proportions to find x and y. Continued Now write proportions to find x and y. To find x: Similarity proportion Cross products Multiply. Divide each side by 4. Example 2-3b

Similarity proportion Continued To find y: Similarity proportion Cross products Multiply. Subtract 6 from each side. Divide each side by 6 and simplify. Example 2-3c

Continued Answer: Example 2-3d

The two polygons are similar. Write a similarity statement. Then find a, b, and ZO. The two polygons are similar. Answer: ; Example 2-3g

Find the perimeter given the figures are similar. Example 4 Find the perimeter given the figures are similar. Answer:

1. Find the ratio of the perimeters. 2. Find the perimeter of EFGH. 6 9 2 3 = 1. Find the ratio of the perimeters. 2. Find the perimeter of EFGH. PABCD = 20 B 6 F 9 4 A A E C 7 3 D G H PEFGH = 30   Answer: The ratio of the perimeters is . The perimeter of EFGH = 30.

C Example 5 If and find the perimeter of Let x represent the perimeter of The perimeter of is: Answer: The perimeter of Example 5-1a

If and RX = 20, find the perimeter of Answer: Example 5-1c

Given that ∆LMN:∆QRT, find the perimeter P and area A of ∆QRS. Example 6: Using Ratios to Find Perimeters and Areas Given that ∆LMN:∆QRT, find the perimeter P and area A of ∆QRS. Perimeter Area 13P = 36(9.1) 132A = (9.1)2(60) P = 25.2 A = 29.4 cm2 The perimeter of ∆QRS is 25.2 cm, and the area is 29.4 cm2.

∆ABC ~ ∆DEF. Find the perimeter and area of ∆ABC. The perimeter of ∆ABC is 27 in., and the area is 31.5 cm2.

Kahoot!

Lesson Summary: Objective: The students will solve problems involving Ratios in Similar Polygons.

Preview of the Next Lesson: Objective: The students will be able to solve problems and prove triangles similar by using AA~.

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