Similar Polygons Section 7-2

Objective Identify similar polygons.

Key Vocabulary Similar polygons Similarity ratio Scale factor

Theorems 7.1 Perimeters of Similar Polygons

What is Similarity? Not Similar Similar Not Similar Similar Triangles

Similarity similar figures Figures that have the same shape but not necessarily the same size are similar figures. But what does “same shape mean”? Are the two heads similar? NOT Similar

Similarity enlargementsreductions Similar shapes can be thought of as enlargements or reductions with no irregular distortions. – So two shapes are similar if one can be enlarged or reduced so that it is congruent to the original.

Similar Polygons When polygons have the same shape but may be different in size, they are called similar polygons. We express similarity using the symbol, ~. (i.e. ΔABC ~ ΔPRS)

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

Similar Polygons The order of the vertices in a similarity statement is very important. It identifies the corresponding angles and sides of the polygons. ΔABC ~ ΔPRS  A   P,  B   R,  C   S AB = BC = CA PR RS SP

Similar Polygons similar polygons Two polygons are similar polygons iff the corresponding angles are congruent and the corresponding sides are proportional. Similarity Statement: Corresponding Angles: Statement of Proportionality:

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

Similarity and Congruence If two polygons are congruent, they are also similar. All of the corresponding angles are congruent, and the lengths of the corresponding sides have a ratio of 1:1.

Example 1 If ΔABC ~ ΔRST, list all pairs of congruent angles and write a proportion that relates the corresponding sides.

Example 1 Use the similarity statement. ΔABC ~ ΔRST Congruent Angles:  A   R,  B   S,  C   T Answer:

Your Turn: If ΔGHK ~ ΔPQR, determine which of the following similarity statements is not true. A.HK ~ QR B. C.  K ~  R D.  GHK ~  QPR

Example 2  PRQ ~  STU. List all pairs of congruent angles.a. Write the ratios of the corresponding sides in a statement of proportionality. b. Check that the ratios of corresponding sides are equal. c. SOLUTION  P   S,  R   T, and  Q   U. a. PR ST RQ TU QP US = = b. c. PR ST = =, RQ TU = =, and QP US = =. The ratios of corresponding sides are all equal to 5 4.

Example 3 Determine whether the triangles are similar. If they are similar, write a similarity statement and find the scale factor of Figure B to Figure A. SOLUTION Check whether the corresponding angles are congruent. 1. From the diagram, you can see that  G   M,  H   K, and  J   L. Therefore, the corresponding angles are congruent.

Example 3 GH MK ÷ 3 12 ÷ 3 = = 3 4 = HJ KL ÷ 4 16 ÷ 4 = = 3 4 = JG LM ÷ 5 20 ÷ 5 = = 3 4 = All three ratios are equal, so the corresponding side lengths are proportional. ANSWER By definition, the triangles are similar.  GHJ ~  MKL. The scale factor of Figure B to Figure A is 3 4. Check whether the corresponding side lengths are proportional. 2.

Your Turn: Determine whether the polygons are similar. If they are similar, write a similarity statement and find the scale factor of Figure B to Figure A ANSWER yes;  XYZ ~  DEF ; 2 3 ANSWER no ≠

Example 4a A. MENUS Tan is designing a new menu for the restaurant where he works. Determine whether the size for the new menu is similar to the original menu. If so, write the similarity statement and scale factor. Explain your reasoning. Original Menu:New Menu:

Example 4a Step 1Compare corresponding angles. Since all angles of a rectangle are right angles and right angles are congruent, corresponding angles are congruent. Step 2Compare corresponding sides. Original New

Example 4a Answer: Since corresponding sides are not proportional, ABCD is not similar to FGHK. So, the menus are not similar.

Example 4b B. MENUS Tan is designing a new menu for the restaurant where he works. Determine whether the size for the new menu is similar to the original menu. If so, write the similarity statement and scale factor. Explain your reasoning. Original Menu:New Menu:

Example 4b Step 1Compare corresponding angles. Since all angles of a rectangle are right angles and right angles are congruent, corresponding angles are congruent. Step 2Compare corresponding sides. Original New

Example 4b Answer: Since corresponding sides are proportional, ABCD ~ RSTU. So the menus are similar with a scale factor of. __ 4 5

Your Turn: A. Thalia is a wedding planner who is making invitations. Determine whether the size for the new invitations is similar to the original invitations used. If so, choose the correct similarity statement and scale factor. A.BCDE ~ FGHI, scale factor = B.BCDE ~ FGHI, scale factor = C.BCDE ~ FGHI, scale factor = D.BCDE is not similar to FGHI. Original:New:

Your Turn: B. Thalia is a wedding planner who is making invitations. Determine whether the size for the new invitations is similar to the original invitations used. If so, choose the correct similarity statement and scale factor. A.BCDE ~ WXYZ, scale factor = B.BCDE ~ WXYZ, scale factor = C.BCDE ~ WXYZ, scale factor = D.BCDE is not similar to WXYZ. __ Original:New:

Example 5a A. The two polygons are similar. Find x. Use the congruent angles to write the corresponding vertices in order. polygon ABCDE ~ polygon RSTUV

Example 5a Write proportions to find x. Similarity proportion Cross Products Property Multiply. Divide each side by 4. Answer: x = __ 9 2

Example 5b B. The two polygons are similar. Find y. Use the congruent angles to write the corresponding vertices in order. polygon ABCDE ~ polygon RSTUV

Example 5b Similarity proportion Cross Products Property Multiply. Subtract 6 from each side. Divide each side by 6 and simplify. AB = 6, RS = 4, DE = 8, UV = y + 1 Answer: y = __ 3 13

Your Turn: A.a = 1.4 B.a = 3.75 C.a = 2.4 D.a = 2 A. The two polygons are similar. Solve for a.

Your Turn: A.1.2 B.2.1 C.7.2 D.9.3 B. The two polygons are similar. Solve for b.

Identifying Similar Triangles When only two congruent angles of a triangle are given, remember that you can use the Third Angles Theorem to establish that the remaining corresponding angles are also congruent. Example:

Example 6 T Thus, all the corresponding angles are congruent. Determine whether the pair of figures is similar. Justify your answer.

Example 6 Now determine whether corresponding sides are proportional. The ratios of the measures of the corresponding sides are equal. Answer: The ratio of the measures of the corresponding sides are equal and the corresponding angles are congruent, so T

Your Turn: Determine whether the pair of figures is similar. Justify your answer. a.

Your Turn: The ratio of the measures of the corresponding sides are equal and the corresponding angles are congruent, Answer: Both triangles are isosceles with base angles measuring 76º and vertex angles measuring 28º.

Your Turn: Answer: Only one pair of angles are congruent, so the triangles are not similar. b. Determine whether the pair of figures is similar. Justify your answer.

Scale Factor scale factor In similar polygons, the ratio of two corresponding sides is called a scale factor. The scale factor depends on the order of comparison. What is the scale factor of the similar polygons shown?

Scale Factor The scale factor between two similar polygons is sometimes called the similarity ratio. Scale factors are usually given for models of real-life objects.

Example 7 An architect prepared a 12-inch model of a skyscraper to look like a real 1100-foot building. What is the scale factor of the model compared to the real building? Before finding the scale factor you must make sure that both measurements use the same unit of measure. 1100(12) = 13,200 inches Scale factor

Example 7 Answer: The ratio comparing the two heights is The scale factor is, which means that the model is the height of the real skyscraper.

Your Turn: A space shuttle is about 122 feet in length. The Science Club plans to make a model of the space shuttle with a length of 24 inches. What is the scale factor of the model compared to the real space shuttle? Answer:

Example 8 The two polygons are similar. Find the scale factor of polygon ABCDE to polygon RSTUV.

Example 8 The scale factor is the ratio of the lengths of any two corresponding sides. Answer:

a. Write a similarity statement. Then find a, b, and ZO. b. Find the scale factor of polygon TRAP to polygon. Answer: The two polygons are similar. Answer: ; Your Turn:

Rectangle WXYZ is similar to rectangle PQRS with a scale factor of 1.5. If the length and width of rectangle PQRS are 10 meters and 4 meters, respectively, what are the length and width of rectangle WXYZ? Write proportions for finding side measures. Let one long side of each WXYZ and PQRS be and one short side of each WXYZ and PQRS be Example 9

Answer:

Your Turn: Quadrilateral GCDE is similar to quadrilateral JKLM with a scale factor of If two of the sides of GCDE measure 7 inches and 14 inches, what are the lengths of the corresponding sides of JKLM? Answer: 5 in., 10 in.

Example 10 The scale on the map of a city is inch equals 2 miles. On the map, the width of the city at its widest point is inches. The city hosts a bicycle race across town at its widest point. Tashawna bikes at 10 miles per hour. How long will it take her to complete the race? Every equals 2 miles. The distance across the city at its widest point is

Example 10 Solve Cross products Divide each side by The distance across the city is 30 miles. Create a proportion relating the measurements to the scale to find the distance in miles. Then use the formula to find the time.

Example 10 Divide each side by 10. Answer: 3 hours It would take Tashawna 3 hours to bike across town.

Your Turn: An historic train ride is planned between two landmarks on the Lewis and Clark Trail. The scale on a map that includes the two landmarks is 3 centimeters = 125 miles. The distance between the two landmarks on the map is 1.5 centimeters. If the train travels at an average rate of 50 miles per hour, how long will the trip between the landmarks take? Answer: 1.25 hours

Perimeters of Similar Polygons In similar polygons, the ratio of any two corresponding lengths is proportional to the scale factor between them. This leads to the following theorem about the perimeters of two similar polygons.

Theorem Perimeters of Similar Polygons If two polygons are similar, then their perimeters are proportional to the scale factor between them.

Example 11 If ABCDE ~ RSTUV, find the scale factor of ABCDE to RSTUV and the perimeter of each polygon.

Example 11 The scale factor ABCDE to RSTUV is or. ___ Write a proportion to find the length of DC. Since DC  AB and AE  DE, the perimeter of ABCDE is or 26. Write a proportion. 4(10.5)= 7 ● DCCross Products Property 6= DCDivide each side by 7.

Example 11 Use the perimeter of ABCDE and scale factor to write a proportion. Let x represent the perimeter of RSTUV. Theorem 7.1 Substitution 4x= (26)(7)Cross Products Property x= 45.5Solve.

Example 11 Answer: The perimeter of ABCDE is 26 and the perimeter of RSTUV is 45.5.

Your Turn: A.LMNOP = 40, VWXYZ = 30 B.LMNOP = 32, VWXYZ = 24 C.LMNOP = 45, VWXYZ = 40 D.LMNOP = 60, VWXYZ = 45 If LMNOP ~ VWXYZ, find the perimeter of each polygon.

Example 12  RST ~  GHJ. Find the value of x. SOLUTION Because the triangles are similar, the corresponding side lengths are proportional. To find the value of x, you can use the following proportion. RS GH TR JG = Write proportion x 9 = Substitute 15 for GH, 10 for RS, 9 for JG, and x for TR. Cross product property 15 · x = 10 · 9

Example x = Divide each side by x = 90 Multiply. Simplify. x = 6

Example 13 The outlines of a pool and the patio around the pool are similar rectangles. Find the ratio of the length of the patio to the length of the pool. a. Find the ratio of the perimeter of the patio to the perimeter of the pool. b. The ratio of the length of the patio to the length of the pool is a. SOLUTION length of pool length of the patio 32 feet 48 feet = 32 ÷ ÷ 16 = =. 2 3

Example 13 perimeter of pool perimeter of patio 96 feet 144 feet = 96 ÷ ÷ 48 = =. 2 3 b.The perimeter of the patio is 2(24) + 2(48) = 144 feet. The perimeter of the pool is 2(16) + 2(32) = 96 feet. The ratio of the perimeter of the patio to the perimeter of the pool is

Your Turn: ANSWER 18 ANSWER 2 1 Find the ratio of the perimeter of  STU to the perimeter of  PQR Find the value of x. In the diagram,  PQR ~  STU.

Assignment Pg. 368 – 371: #1 – 37 odd