Use Similar Polygons VOCABULARY. Similar polygons. Scale factor of two similar polygons. a. List all pairs of congruent angles.

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1 6.3 Use Similar olygons Goal p Use proportions to identify similar polygons. Your Notes VOCABUAY Similar polygons Scale factor of two similar polygons In a statement of proportionality, any pair of ratios forms a true proportion. Example 1 Use similarity statements In the diagram, nabc, ndef. B 10 a. ist all pairs of congruent angles. A b. Check that the ratios of corresponding side lengths are equal. c. Write the ratios of the corresponding side lengths in a statement of proportionality. D C E 1 1 F a. A >, B >, C > b. AB DE BC EF CA FD c. The ratios in part (b) are equal, so. Checkpoint Complete the following exercise. 1. Given nq, nxyz, list all pairs of congruent angles. Write the ratios of the corresponding side lengths in a statement of proportionality. Copyright Holt McDougal. All rights reserved. esson 6.3 Geometry Notetaking Guide 13

2 6.3 Use Similar olygons Goal p Use proportions to identify similar polygons. Your Notes VOCABUAY Similar polygons Two polygons are similar polygons if corresponding angles are congruent and corresponding side lengths are proportional. Scale factor of two similar polygons If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the scale factor. In a statement of proportionality, any pair of ratios forms a true proportion. Example 1 Use similarity statements In the diagram, nabc, ndef. B 10 a. ist all pairs of congruent angles. A b. Check that the ratios of corresponding side lengths are equal. c. Write the ratios of the corresponding side lengths in a statement of proportionality. D C E 1 1 F a. A > D, B > E, C > F b. AB DE CA FD BC EF 2 3 c. The ratios in part (b) are equal, so AB DE BC EF CA FD. Checkpoint Complete the following exercise. 1. Given nq, nxyz, list all pairs of congruent angles. Write the ratios of the corresponding side lengths in a statement of proportionality. > X, Q > Y, > Z; Q XY Q YZ ZX Copyright Holt McDougal. All rights reserved. esson 6.3 Geometry Notetaking Guide 13

3 Example 2 Find the scale factor Determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor of ABCD to JKM. M Step 1 Identify pairs of congruent angles. From the diagram, you can see that B >, C >, and D >. Angles and are right angles, so >. So, the corresponding angles are. Step 2 Show that corresponding side lengths are proportional. AB JK BC K CD M A D B C 21 J AD JM The ratios are equal, so the corresponding side lengths are. So ABCD,. The scale factor of ABCD to JKM is. K Example 3 Use similar polygons In the diagram, nbcd, nst. Find the value of x. The triangles are similar, so the corresponding side lengths are. C S x There are several ways to write the proportion. For example, you could write BD T CD ST. BC x ST x Write proportion. Substitute. Cross roducts roperty B D 10 T x Solve for x. 14 esson 6.3 Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.

4 Example 2 Find the scale factor Determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor of ABCD to JKM. M Step 1 Identify pairs of congruent angles. From the diagram, you can see that B > K, C >, and D > M. Angles A and J are right angles, so A > J. So, the corresponding angles are congruent. Step 2 Show that corresponding side lengths are proportional. AB JK 4 7 CD M 4 7 A D C B 21 J BC K 4 7 AD JM The ratios are equal, so the corresponding side lengths are proportional. So ABCD, JKM. The scale factor of ABCD to JKM is 4 7. K Example 3 Use similar polygons In the diagram, nbcd, nst. Find the value of x. There are several ways to write the proportion. For example, you could write BD T CD ST. The triangles are similar, so the corresponding side lengths are proportional. BC S CD ST x x 3 Write proportion. Substitute. Cross roducts roperty B C 13 D 24 S 10 x T x 26 Solve for x. 14 esson 6.3 Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.

5 Checkpoint In the diagram, FGHJ, MN. 2. What is the scale factor of MN to FGHJ? 3. Find the value of x. G 1 F M 16 x J 1 H N THEOEM 6.1: EIMETES OF SIMIA OYGONS If two polygons are similar, then the ratio of their perimeters is equal to the Q ratios of their corresponding K side lengths. If KMN, QS, then M S K 1 M 1 MN 1 NK Q 1 Q 1 S 1 S. N Example 4 Basketball A larger cement court is being poured for a basketball hoop in place of a smaller one. The court will be 20 feet wide and 2 feet long. The old court was similar in shape, but only 16 feet wide. a. Find the scale factor of the new court to the old court. b. Find the perimeters of the new court and the old court. a. Because the new court will be similar to the old court, the scale factor is the ratio of the widths,. b. The new court s perimeter is feet. Use Theorem 6.1 to find the perimeter x of the old court. 90 x Find perimeters of similar figures Use Theorem 6.1 to write a proportion. x Simplify. The perimeter of the old court was feet. Copyright Holt McDougal. All rights reserved. esson 6.3 Geometry Notetaking Guide 1

6 Checkpoint In the diagram, FGHJ, MN. 2. What is the scale factor of MN to FGHJ? 4 3. Find the value of x F 20 J G 1 40 H M x 16 N THEOEM 6.1: EIMETES OF SIMIA OYGONS If two polygons are similar, then the ratio of their perimeters is equal to the Q ratios of their corresponding K side lengths. If KMN, QS, then M S K 1 M 1 MN 1 NK Q 1 Q 1 S 1 S K Q M Q N MN S NK S. Example 4 Basketball A larger cement court is being poured for a basketball hoop in place of a smaller one. The court will be 20 feet wide and 2 feet long. The old court was similar in shape, but only 16 feet wide. a. Find the scale factor of the new court to the old court. b. Find the perimeters of the new court and the old court. a. Because the new court will be similar to the old court, the scale factor is the ratio of the widths, b. The new court s perimeter is 2(20) 1 2(2) 90 feet. Use Theorem 6.1 to find the perimeter x of the old court. 90 x 4 Find perimeters of similar figures Use Theorem 6.1 to write a proportion. x 72 Simplify. The perimeter of the old court was 72 feet. Copyright Holt McDougal. All rights reserved. esson 6.3 Geometry Notetaking Guide 1

7 COESONDING ENGTHS IN SIMIA OYGONS If two polygons are similar, then the ratio of any two corresponding lengths in the polygons is equal to the of the similar polygons. Example Use a scale factor In the diagram, nfgh, njgk. Find the length of the altitude G. G First, find the scale factor of nfgh to njgk. J M K FH F H Because the ratio of the lengths of the altitudes in similar triangles is equal to the scale factor, you can write the following proportion. G GM Write proportion. G Substitute for GM. G Multiply each side by and simplify. The length of altitude G is. Checkpoint In the diagrams, nq, nwxy. 4. Find the perimeter of nwxy. Q X 40 W Y Homework. Find the length of median QS. X Q W Z Y S esson 6.3 Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.

8 COESONDING ENGTHS IN SIMIA OYGONS 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. Example Use a scale factor In the diagram, nfgh, njgk. Find the length of the altitude G. G First, find the scale factor of nfgh to njgk. FH JK Because the ratio of the lengths of the altitudes in similar triangles is equal to the scale factor, you can write the following proportion. G GM Write proportion. G Substitute for GM. G 22.4 Multiply each side by and simplify. The length of altitude G is F J M K H Checkpoint In the diagrams, nq, nwxy. 4. Find the perimeter of nwxy. Q The perimeter of nwxy is 0. X 40 W 72 Y 63 1 Homework. Find the length of median QS. QS 26 W X 39 Z 24 Y Q S esson 6.3 Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.

9 Focus On atterns Use after esson 6.3 Your Notes Fibonacci Sequence and the Golden atio Goal p Use the Fibonacci sequence and the golden ratio. VOCABUAY Fibonacci sequence Golden ratio Golden rectangle Example 1 Find the eighth and ninth terms of the Fibonacci sequence. Each term of the Fibonacci sequence after the second term is the of the previoue terms. 3 rd term Sum of 1 st and 2 nd terms 4 th term Sum of 2 nd and 3 rd terms th term 2 1 Sum of 3 rd and terms 6 th term 1 Sum of and th terms 7 th term 1 Sum of th and terms th term Sum of and terms 9 th term Find terms in the Fibonacci sequence Checkpoint Find terms of the Fibonacci sequence. 1. What are the tenth and eleventh terms of the Fibonacci sequence? Copyright Holt McDougal. All rights reserved. 6.3 Foucs on atterns Geometry Notetaking Guide 17

10 Focus On atterns Use after esson 6.3 Your Notes Fibonacci Sequence and the Golden atio Goal p Use the Fibonacci sequence and the golden ratio. VOCABUAY Fibonacci sequence The sequence 1, 1, 2, 3,,..., which uses the rule that each term after the second term is the sum of the previous two terms Golden ratio The irrational number approached by the ratios of consecutive terms from the Fibonacci sequence. Golden rectangle A rectangle whose sides form a golden ratio. Example 1 Find terms in the Fibonacci sequence Find the eighth and ninth terms of the Fibonacci sequence. Each term of the Fibonacci sequence after the second term is the sum of the previoue two terms. 3 rd term Sum of 1 st and 2 nd terms 4 th term Sum of 2 nd and 3 rd terms th term Sum of 3 rd and 4 th terms 6 th term 3 1 Sum of 4 th and th terms 7 th term 1 13 Sum of th and 6 th terms th term Sum of 6 th and 7 th terms 9 th term Sum of 7 th and th terms Checkpoint Find terms of the Fibonacci sequence. 1. What are the tenth and eleventh terms of the Fibonacci sequence? ; 9 Copyright Holt McDougal. All rights reserved. 6.3 Foucs on atterns Geometry Notetaking Guide 17

11 Example 2 Find ratios of terms in the Fibonacci sequence Find the ratios of consecutive terms in the Fibonacci sequence using the seventh, eighth, and ninth terms. ound to the nearest The seventh, eighth, and ninth terms of the Fibonacci sequence are,, and. th term 13 Checkpoint Complete the following exercise. 2. Which two consecutive terms of the Fibonacci sequence will give you a ratio of approximately 1.61? Example 3 The golden rectangle Show that the figure is nearly a golden rectangle. 6. feet For a rectangle to be a golden rectangle, 4 feet Homework length width 1 length For the figure shown, , and , the figure nearly a golden rectangle. Checkpoint Complete the following exercise. 3. Is a 1 m 3 13 m rectangle nearly a golden rectangle? Focus on atterns Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.

12 Example 2 Find ratios of terms in the Fibonacci sequence Find the ratios of consecutive terms in the Fibonacci sequence using the seventh, eighth, and ninth terms. ound to the nearest The seventh, eighth, and ninth terms of the Fibonacci sequence are 13, 21, and 34. th term 7th term th term th term Checkpoint Complete the following exercise. 2. Which two consecutive terms of the Fibonacci sequence will give you a ratio of approximately 1.61? ninth and tenth terms Example 3 The golden rectangle Show that the figure is nearly a golden rectangle. 6. feet Homework For a rectangle to be a golden rectangle, length width width 1 length length For the figure shown, , and Yes, the figure is nearly a golden rectangle. Checkpoint Complete the following exercise. 3. Is a 1 m 3 13 m rectangle nearly a golden rectangle? No 4 feet Focus on atterns Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.