Ratio of a to b If a and b are two numbers or quantities and b Þ 0, then the ratio of a to b is a }

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1 6.1 Ratios, Proportions, and the Geometric Mean Goal p Solve problems by writing and solving proportions. Your Notes VOULRY Ratio of a to b If a and b are two numbers or quantities and b Þ 0, then the ratio of a to b is a b. Proportion n equation that states that two ratios are equal is a proportion. Means, extremes In the proportion a b c d, b and c are the means, and a and d are the extremes. Geometric mean The geometric mean of two positive numbers a and b is the positive number x that satisfies a x x b. Example 1 Simplify ratios Simplify the ratio. (See Table of Measures, p. 921) a. 76 cm : 8 cm b. 4 ft 24 in. For help with conversion factors, see p cm a. Write 76 cm : 8 cm as units and simplify. 8 cm 76 cm : 2 8 cm 2. Then divide out the b. To simplify a ratio with unlike units, multiply by a conversion factor. 4 ft 24 in. 4 ft 24 in. p 12 in. 1 ft Lesson 6.1 Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.

2 Example 2 Use a ratio to find a dimension Painting You are painting barn doors. You know that the perimeter of the doors is 64 feet and that the ratio of the length to the height is 3 :. Find the area of the doors. Step 1 Write expressions for the length and height. ecause the ratio of the length to height is 3 :, you can represent the length by 3 x and the height by x. Step 2 Solve an equation to find x. 2l 1 2w P Formula for perimeter 2( 3 x) 1 2( x) 64 Substitute. 16 x 64 Multiply and combine like terms. x 4 Divide each side by 16. Step 3 Evaluate the expressions for the length and height. Substitute the value of x into each expression. Length: 3 x 3 ( 4 ) 12 Height: x ( 4 ) 20 The doors are 12 feet long and 20 feet high, so the area is 12 p ft 2. heckpoint In Exercises 1 and 2, simplify the ratio meters to 18 meters yd : 9 ft 2 to 9 11 : 1 3. The perimeter of a rectangular table is 21 feet and the ratio of its length to its width is : 2. Find the length and width of the table. length: 7. feet, width: 3 feet opyright Holt McDougal. ll rights reserved. Lesson 6.1 Geometry Notetaking Guide 14

3 Example 3 Use extended ratios The measures of the angles in nd are in the extended ratio of 2:3:4. Find the measures of the angles. egin by sketching the triangle. Then use the extended ratio of 2:3:4 to label the measures as 2 x8, 3 x8, and 4 x8. 2 x8 1 3 x8 1 4 x Triangle Sum Theorem 9 x 180 ombine like terms. x 20 Divide each side by 9. The angle measures are 2( 208 ) 408, 3( 208 ) 608, and 4( 208 ) x8 2x8 4x8 heckpoint omplete the following exercise. 4. triangle s angle measures are in the extended ratio of 1:4:. Find the measures of the angles. 188, 728, 908 PROPERTY OF PROPORTIONS 1. ross Products Property In a proportion, the product of the extremes equals the product of the means. If a b c where b Þ 0 and d Þ 0, then ad bc. d p p Lesson 6.1 Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.

4 Example 4 Solve proportions Solve the proportion. In part (a), you could multiply each side by the denominator, 16. Then 16 p p x 16 so 12 x. a. b. 3 4 x 16 Original proportion 3 p 16 4 p x ross Products Property 48 4 x Multiply. 12 x Divide each side by 4. 3 x x Original proportion 3 p x 2 (x 1 1) ross Products Property 3 x 2 x 1 2 Distributive Property x 2 Subtract 2x from each side. Example Solve a real-world problem owling You want to find the total number of rows of boards that make up 24 lanes at a bowling alley. You know that there are 117 rows in 3 lanes. Find the total number of rows of boards that make up the 24 lanes. Write and solve a proportion involving two ratios that compare the number of rows with the number of lanes n 24 number of rows number of lanes Write proportion. 117 p 24 3 p n ross Products Property 936 n Simplify. There are 936 rows of boards that make up the 24 lanes. GEOMETRI MEN The geometric mean of two positive numbers a and b is the positive number x that satisfies a x x b. So, x 2 ab and x Ï ab. opyright Holt McDougal. ll rights reserved. Lesson 6.1 Geometry Notetaking Guide 147

5 Example 6 Find a geometric mean Find the geometric mean of 16 and 48. x Ï ab Definition of geometric mean Ï 16 p 48 Substitute 16 for a and 48 for b. Ï 16 p 16 p 3 Factor. 16 Ï 3 Simplify. The geometric mean of 16 and 48 is 16 Ï 3 ø heckpoint omplete the following exercises.. Solve 8 y 2 x Solve 3 y 20 x 9 2x small gymnasium contains 10 sets of bleachers. You count 192 spectators in 3 sets of bleachers and the spectators seem to be evenly distributed. Estimate the total number of spectators. about 640 spectators Homework 8. Find the geometric mean of 14 and Ï 14 ø Lesson 6.1 Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.

6 6.2 Use Proportions to Solve Geometry Problems Goal p Use proportions to solve geometry problems. Your Notes VOULRY Scale drawing scale drawing is a drawing that is the same shape as the object it represents. Scale The scale is a ratio that describes how the dimensions in the drawing are related to the actual dimensions of the object. DDITIONL PROPERTIES OF PROPORTIONS 2. Reciprocal Property If two ratios are equal, then their reciprocals are also equal. If a b c d, then b a d c. 3. If you interchange the means of a proportion, then you form another true proportion. If a b c d, then a c b d. 4. In a proportion, if you add the value of each ratio s denominator to its numerator, then you form another true proportion. If a b c d, then a 1 b. b c 1 d d opyright Holt McDougal. ll rights reserved. Lesson 6.2 Geometry Notetaking Guide 149

7 Example 1 Use properties of proportions In the diagram, DF EF. Write four true proportions. ecause DF 12, then EF 18 9 x. E Reciprocal Property: The reciprocals are equal, so x 9. Property 3: You can interchange the means, so x D 18 F x Property 4: You can add the denominators to the 30 numerators, so x. x Example 2 In the diagram, JL LH JK KG. Find JH and JL. JL 1 LH LH JL LH JK KG Use proportions with geometric figures JK 1 KG KG x x 2(1 1 ) G K Given x 8 Solve for x. So JH 8 and JL J x L 2 H Property of Proportions (Property 4) Substitution Property of Equality ross Products Property heckpoint omplete the following exercises. 1. In Example 1, find 2. In Example 2, the value of x. KL GH JK. Find GH. JG x 13. GH Lesson 6.2 Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.

8 Example 3 Find the scale of a drawing Keys The length of the key in the scale drawing is 7 centimeters. The length of the actual key is 4 centimeters. What is the scale of the drawing? To find the scale, write the ratio of a length in the drawing to an actual length, then rewrite the ratio so that the denominator is 1. length in drawing 7 cm 4 cm length of key The scale of the drawing is 1.7 cm : 1 cm. heckpoint omplete the following exercise. 3. In Example 3, suppose the length of the key in the scale drawing is 6 centimeters. Find the new scale of the drawing. 1. cm : 1 cm Example 4 Use a scale drawing Maps The scale of the map at the right is 1 inch : 8 miles. Find the actual distance from Westbrook to ooley. Use a ruler. The distance from Westbrook to ooley on the map is about 1.2 inches. Let x be the actual distance in miles. 1.2 in. x mi 1 in. 8 mi x 1.2(8) distance on map actual distance Greenbow ross Products Property ooley Westbrook Jackson x 10 Simplify. The actual distance from Westbrook to ooley is about 10 miles. opyright Holt McDougal. ll rights reserved. Lesson 6.2 Geometry Notetaking Guide 11

9 Example Solve a multi-step problem Scale Model You buy a 3-D scale model of the Sunsphere in Knoxville, TN. The actual building is 266 feet tall. Your model is 20 inches tall, and the diameter of the dome on your scale model is about.6 inches. a. What is the diameter of the actual dome? b. How many times as tall as your model is the actual building? a. 20 in. 266 ft.6 in. x ft measurement on model measurement on actual building 20 x ross Products Property x ø 74. Divide each side by 20. The diameter of the actual dome is about 74. feet. b. To simplify a ratio with unlike units, multiply by a conversion factor. 266 ft 20 in. 266 ft 20 in. p 12 in. 1 ft 19.6 The actual building is 19.6 times as tall as the model. heckpoint omplete the following exercises. 4. Two landmarks are 130 miles from each other. The landmarks are 6. inches apart on a map. Find the scale of the map. 1 inch : 20 miles Homework. Your friend has a model of the Sunsphere that is inches tall. What is the approximate diameter of the dome on your friend s model? about 1.4 inches 12 Lesson 6.2 Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.

10 6.3 Use Similar Polygons Goal p Use proportions to identify similar polygons. Your Notes VOULRY 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, n, ndef a. List all pairs of congruent angles. b. heck that the ratios of corresponding side lengths are equal. c. Write the ratios of the corresponding side lengths in a statement of proportionality. 12 D E F a. > D, > E, > F b. DE FD EF c. The ratios in part (b) are equal, so DE EF FD. heckpoint omplete the following exercise. 1. Given npqr, nxyz, list all pairs of congruent angles. Write the ratios of the corresponding side lengths in a statement of proportionality. P > X, Q > Y, R > Z; PQ XY QR YZ RP ZX opyright Holt McDougal. ll rights reserved. Lesson 6.3 Geometry Notetaking Guide 13

11 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 D to JKLM. 14 M Step 1 Identify pairs of congruent angles. From the diagram, you can see that > K, > L, and D > M. ngles and J are right angles, so > J. So, the corresponding angles are congruent. Step 2 Show that corresponding side lengths are proportional. JK D LM D J 14 KL D JM The ratios are equal, so the corresponding side lengths are proportional. So D, JKLM. The scale factor of D to JKLM is 4 7. L K 14 Example 3 Use similar polygons In the diagram, nd, nrst. Find the value of x. There are several ways to write the proportion. For example, you could write D RT D ST. The triangles are similar, so the corresponding side lengths are proportional. RS D ST x 12x 312 Write proportion. Substitute. ross Products Property D 24 R S 10 x T x 26 Solve for x. 14 Lesson 6.3 Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.

12 heckpoint In the diagram, FGHJ, LMNP. 2. What is the scale factor of LMNP to FGHJ? 4 3. Find the value of x F 20 J G 1 40 H M 12 L x 16 P 12 N THEOREM 6.1: PERIMETERS OF SIMILR POLYGONS If two polygons are similar, then the ratio of their perimeters is equal to the Q P ratios of their corresponding L K side lengths. If KLMN, PQRS, then M S R KL 1 LM 1 MN 1 NK PQ 1 QR 1 RS 1 SP KL PQ LM QR N MN RS NK SP. Example 4 asketball 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. ecause 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. opyright Holt McDougal. ll rights reserved. Lesson 6.3 Geometry Notetaking Guide 1

13 ORRESPONDING LENGTHS IN SIMILR POLYGONS 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 GL. G First, find the scale factor of nfgh to njgk. FH JK ecause the ratio of the lengths of the altitudes in similar triangles is equal to the scale factor, you can write the following proportion. GL GM 8 Write proportion. GL 8 Substitute 14 for GM. 14 GL 22.4 Multiply each side by 14 and simplify. The length of altitude GL is F J 8 14 L M 8 K H heckpoint In the diagrams, npqr, nwxy. 4. Find the perimeter of nwxy. Q The perimeter of nwxy is 120. X 40 W 72 Y P R Homework. Find the length of median QS. QS 26 W X 39 Z 24 P Y Q S 16 R 16 Lesson 6.3 Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.

14 Focus On Patterns Use after Lesson 6.3 Your Notes Fibonacci Sequence and the Golden Ratio Goal p Use the Fibonacci sequence and the golden ratio. VOULRY 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 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 Sum of 4 th and th terms 7 th term Sum of th and 6 th terms 8 th term Sum of 6 th and 7 th terms 9 th term Sum of 7 th and 8 th terms heckpoint Find terms of the Fibonacci sequence. 1. What are the tenth and eleventh terms of the Fibonacci sequence? ; 89 opyright Holt McDougal. ll rights reserved. 6.3 Foucs on Patterns Geometry Notetaking Guide 17

15 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. Round to the nearest The seventh, eighth, and ninth terms of the Fibonacci sequence are 13, 21, and 34. 8th term 7th term th term 8th term heckpoint omplete the following exercise. 2. Which two consecutive terms of the Fibonacci sequence will give you a ratio of approximately 1.618? 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. heckpoint omplete the following exercise. 3. Is a 1 m 3 13 m rectangle nearly a golden rectangle? No 4 feet Focus on Patterns Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.

16 6.4 Prove Triangles Similar by Goal p Use the Similarity Postulate. Your Notes POSTULTE 22: NGLE-NGLE () SIMILRITY POSTULTE K L If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. J X Y njkl, nxyz Z Example 1 Determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning. Use the Similarity Postulate E F 688 ecause they are both right angles, and E are congruent. y the Triangle Sum Theorem, m 1808, so m 228. Therefore, and D are congruent. So, n, ndef by the Similarity Postulate. 228 D heckpoint Determine whether the triangles are similar. If they are, write a similarity statement. 1. P R G P 278 G 1108 R 38 Q H 278 Q F 788 H F nfgh, nrqp not similar opyright Holt McDougal. ll rights reserved. Lesson 6.4 Geometry Notetaking Guide 19

17 Example 2 Show that triangles are similar Show that the two triangles are similar. a. nrtv and nrqs b. nlmn and nnop T 498 R V M O Q 498 S a. You may find it helpful to redraw the triangles separately. ecause m RTV and m Q both equal 498, RTV > Q. y the Reflexive Property, R > R. So, nrtv, nrqs by the Similarity Postulate. b. The diagram shows L > ONP. It also shows that MN i OP so LNM > P by the orresponding ngles Postulate. So, nlmn, nnop by the Similarity Postulate. L N P heckpoint omplete the following exercise. 3. Show that nd, nefd. F D E ecause they are both right angles, > F. You know that D > FDE by the Vertical ngles ongruence Theorem. So, nd, nefd by the Similarity Postulate. 160 Lesson 6.4 Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.

18 Example 3 Using similar triangles Height lifeguard is standing beside the lifeguard chair on a beach. The lifeguard is 6 feet 4 inches tall and casts a shadow that is 48 inches long. The chair casts a shadow that is 6 feet long. How tall is the chair? The lifeguard and the chair form sides of two right triangles with the ground, as shown below. The sun s rays hit the lifeguard and the chair at the same angle. You have two pairs of congruent angles, so the triangles are similar by the Similarity Postulate. 6 ft 4 in. x ft 48 in. 6 ft You can use a proportion to find the height x. Write 6 feet 4 inches as 76 inches so you can form two ratios of feet to inches. x ft 76 in. 6 ft 48 in. Write proportion of side lengths. 48 x 46 ross Products Property x 9. Solve for x. The chair is 9. feet tall. heckpoint omplete the following exercise. 4. In Example 3, how long is the shadow of a person that is 4 feet 9 inches tall? Homework 3 feet opyright Holt McDougal. ll rights reserved. Lesson 6.4 Geometry Notetaking Guide 161

19 6. Prove Triangles Similar by SSS and SS Goal p Use the SSS and SS Similarity Theorems. Your Notes THEOREM 6.2: SIDE-SIDE-SIDE (SSS) SIMILRITY THEOREM If the corresponding side lengths of two triangles are proportional, then the triangles are similar. R If RS ST, then n, nrst. TR S T Example 1 Use the SSS Similarity Theorem Is either ndef or nghj similar to n? H D 8 F 4 6 E J G When using the SSS Similarity Theorem, compare the shortest sides, the longest sides, and then the remaining sides. ompare n and ndef by finding ratios of corresponding side lengths. Shortest sides Longest sides Remaining sides 4 2 FD EF DE 8 The ratios are not all equal, so n and ndef are not similar. ompare n and nghj by finding ratios of corresponding side lengths. Shortest sides Longest sides Remaining sides GH JG HJ ll the ratios are equal, so n, n GHJ. 162 Lesson 6. Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.

20 Example 2 Find the value of x that makes n, ndef. x D 2x 1 3 F Step 1 Find the value of x that makes corresponding side lengths proportional x Write proportion. 4 p (x 2 3) ross Products Property x 2 30 Simplify. 11 x Solve for x. Step 2 heck that the side lengths are proportional when x 11. x DF 2x EF 0 DE Use the SSS Similarity Theorem EF 0 DF When x 11, the triangles are similar by the SSS Similarity Theorem. E heckpoint omplete the following exercises. 1. Which of the three triangles are similar? npqr, nzxy Q 2. Suppose is not given in n. What value of would make n similar to nqrp? P 4 3 R Y X 27 Z opyright Holt McDougal. ll rights reserved. Lesson 6. Geometry Notetaking Guide 163

21 THEOREM 6.3: SIDE-NGLE-SIDE (SS) SIMILRITY THEOREM If an angle of one triangle is X M congruent to an angle of a second triangle and the lengths P N of the sides including these Z Y angles are proportional, then the triangles are similar. If X > M, and ZX PM XY, then nxyz, nmnp. MN Example 3 Use the SS Similarity Theorem irdfeeder You are drawing a design for a birdfeeder. an you construct the top so it is similar to the bottom using the angle measure and lengths shown? 32 cm 32 cm 878 D oth m and m E equal 878, so 20 cm E > E. Next, compare the ratios of the lengths of the sides that include and E. DE 32 EF 20 8 F 20 cm 878 The lengths of the sides that include and E are proportional. So, by the SS Similarity Theorem, n, ndef. Yes, you can make the top similar to the bottom. heckpoint omplete the following exercise. 3. In Example 3, suppose you use equilateral triangles on the top and bottom. re the top and bottom similar? Explain. Yes, the top and bottom are similar. If the side length of the top is a and the side length of the bottom is b, the ratios of the side lengths are a and the angles are all 608. The triangles are b similar by SS or SSS. 164 Lesson 6. Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.

22 TRINGLE SIMILRITY POSTULTE ND THEOREMS Similarity Postulate If > D and > E, then n, ndef. SSS Similarity Theorem If then n, ndef. DE EF DF, SS Similarity Theorem If > D and DE, then n, ndef. DF To identify corresponding parts, redraw the triangles so that the corresponding parts have the same orientation. P 18 T 24 R Q 9 S 12 R Example 4 Tell what method you would use to show that the triangles are similar. Find the ratios of the lengths of the corresponding sides. Shorter sides Longer sides hoose a method QR RS PR RT P 18 Q 9 The corresponding side lengths are proportional. The included angles PRQ and TRS are congruent because they are vertical angles. So, npqr, ntsr by the SS Similarity Theorem. R S T heckpoint omplete the following exercise. 4. Explain how to show njkl, nlkm. L Homework Show that the corresponding side lengths are proportional, then use the SSS Similarity Theorem to show njkl, nlkm. J K M opyright Holt McDougal. ll rights reserved. Lesson 6. Geometry Notetaking Guide 16

23 6.6 Use Proportionality Theorems Goal p Use proportions with a triangle or parallel lines. Your Notes THEOREM 6.4: TRINGLE PROPORTIONLITY THEOREM Q T If a line parallel to one side of a R triangle intersects the other two S U sides, then it divides the two sides proportionally. If TU i QS, then RT TQ RU US. THEOREM 6.: ONVERSE OF THE TRINGLE PROPORTIONLITY THEOREM If a line divides two sides of a Q T triangle proportionally, then it is parallel to the third side. S U If RT TQ RU US, then TU i QS. R Example 1 In the diagram, QS i UT, RQ 10, RS 12, and ST 6. What is the length of QU? R Find the length of a segment U Q S 6 T RQ QU RS ST Triangle Proportionality Theorem QU 6 Substitute p QU ross Products Property QU Divide each side by Lesson 6.6 Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.

24 Þ Your Notes Example 2 Solve a real-world problem erodynamics spoiler for a remote controlled car is shown where 31 mm, 19 mm, D 27 mm, and DE 23 mm. Explain why D is not parallel to E. Not drawn to scale D E Find and simplify the ratios of lengths determined by D. D DE ecause Þ 31 19, D E. heckpoint omplete the following exercises. 1. Find the length of KL. KL 33 J 18 K L N 24 M Determine whether QT i RS. No; QT is not parallel to RS. P 18 Q 38 R 36 T 80 S opyright Holt McDougal. ll rights reserved. Lesson 6.6 Geometry Notetaking Guide 167

25 THEOREM 6.6 If three parallel lines intersect two transversals, then they divide the transversals proportionally. r s t U W Y V X Z UW WY VX XZ l m THEOREM 6.7 If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. D D D Example 3 Use Theorem 6.6 Farming farmer s land is divided by a newly constructed interstate. The distances shown are in meters. Find the distance between the north border and the south border of the farmer s land. Use Theorem 6.6. DE EF 1 DE 1 EF EF F West order North order E 3000 Interstate South order D Parallel lines divide transversals proportionally. Property of proportions (Property 4) Substitute. Simplify Multiply each side by 2000 and simplify. The distance between the north border and the south border is 4400 meters. 168 Lesson 6.6 Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.

26 Example 4 Use Theorem 6.7 In the diagram, DEG > GEF. Use the given side lengths to find the length of DG. D 8 12 x E G 14 ecause ###$ EG is an angle bisector of DEF, you can apply Theorem 6.7. Let GD x. Then GF 14 2 x. GF GD EF ED ngle bisector divides opposite side proportionally x 12 x 8 Substitute. 12 x x ross Products Property x.6 Solve for x. F heckpoint Find the length of Homework 2 Ï 2 D opyright Holt McDougal. ll rights reserved. Lesson 6.6 Geometry Notetaking Guide 169

27 6.7 Similarity Transformations and oordinate Geometry Goal p Perform dilations. Your Notes VOULRY Dilation dilation is a transformation that stretches or shrinks a figure to create a similar figure. enter of dilation In a dilation, a figure is enlarged or reduced with respect to a fixed point called the center of dilation. Scale factor of a dilation The scale factor k of a dilation is the ratio of a side length of the image to the corresponding side length of the original figure. Reduction dilation where 0 < k < 1 is a reduction. Enlargement dilation where k > 1 is an enlargement. OORDINTE NOTTION FOR DILTION You can describe a dilation with respect to the origin with the notation (x, y) (kx, ky), where k is the scale factor. If 0 < k < 1, the dilation is a reduction. If k > 1, the dilation is an enlargement. 170 Lesson 6.7 Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.

28 ll of the dilations in this lesson are in the coordinate plane and each center of dilation is the origin. Example 1 Draw a dilation with a scale factor greater than 1 Draw a dilation of quadrilateral D with vertices (2, 0), (6, 24), (8, 2), and D(6, 4). Use a scale factor of 1 2. First draw D. Find the dilation of each vertex by multiplying its coordinates by 1. Then draw the dilation. 2 (x, y) x, 1 2 y y D 2 (2, 0) L (1, 0) (6, 24) M (3, 22) (8, 2) N (4, 1) D(6, 4) P (3, 2) 1 L 1 M P N x Example 2 triangle has the vertices (2, 21), (4, 21), and (4, 2). The image of n after a dilation with a scale factor of 2 is ndef. a. Sketch n and ndef. b. Verify that n and ndef are similar. a. The scale factor is greater than 1, so the dilation is an enlargement. (x, y) ( 2 x, 2 y) (2, 21) D(4, 22) (4, 21) E(8, 22) (4, 2) F(8, 4) 1 y 1 b. ecause and E are both right angles, > E. Show that the lengths of the sides that include and E are proportional. DE 0 EF Verify that a figure is similar to its dilation The lengths are proportional. So, n, ndef by the SS Similarity Theorem. D F E x opyright Holt McDougal. ll rights reserved. Lesson 6.7 Geometry Notetaking Guide 171

29 Example 3 Find a scale factor Magnets You are making your own photo magnets. Your photo is 8 inches by 10 inches. The image on the magnet is 2.8 inches by 3. inches. What is the scale factor of the reduction? The scale factor is the ratio of a side length of the magnet image to a side length of the original photo, or the scale factor is in. 8 in.. In simplest form, heckpoint omplete the following exercises. 1. triangle has the vertices (21, 21), (0, 1), and D(1, 0). Find the coordinates of L, M, and N so that nlmn is a dilation of nd with a scale factor of 4. Sketch nd and nlmn. L(24, 24), M(0, 4), N(4, 0) y M 1 D 1 N x L 2. In Example 3, what is the scale factor of the reduction if your photo is 4 inches by inches? Lesson 6.7 Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.

30 Example 4 Find missing coordinates You want to create a quadrilateral JKLM that is similar to quadrilateral PQRS. What are the coordinates of M? Determine if JKLM is a dilation of PQRS by checking whether the same scale factor can be used to obtain J, K, and L from P, Q, and R. (x, y) (kx, ky) P( 0, 3 ) J( 0, 6 ) k 2 Q( 2, 2 ) K( 4, 4 ) k 2 R( 2, 0 ) L( 4, 0 ) k 2 ecause k is the same in each case, the image is a dilation with a scale factor of 2. So, you can use the scale factor to find the image M of point S. S(, ) M( 2 p, 2 p ) M( 10, 10 ) y P 1 J 1 R Q K L S x heckpoint omplete the following exercise. 3. You want to create a quadrilateral QRST that is similar to quadrilateral WXYZ. What are the coordinates of T? y Q Z S Homework W 1 1 Y X R x T(10, 1) opyright Holt McDougal. ll rights reserved. Lesson 6.7 Geometry Notetaking Guide 173

31 Words to Review Give an example of the vocabulary word. Ratio 4 3 or 4 : 3 Means The means of a b and c. b c d are Proportion x Extremes The extremes of a are a and d. b c d Geometric mean The geometric mean of two positive numbers a and b is the positive number x that satisfies a x x b. Scale Scale drawings scale drawing is a drawing that is the same shape as the object it represents. map is an example of a scale drawing. Similar polygons The scale of a map is 1 inch to 2 miles. D F J G H D, FGHJ Scale factor of two similar polygons 3 D F G 2 H J Fibonacci sequence 1,1,2,3,,8,13,... The scale factor of D to FGHJ is 1 2. Golden ratio Words to Review Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.

32 Golden rectangle Dilation y X 1 O Y Z x enter of dilation Scale factor of a dilation y X y X O Y Z x O Y Z x The center of dilation is (0, 0). Reduction dilation with a scale factor greater than 0 and less than 1 is a reduction. The scale factor of the dilation is XY. Enlargement dilation with a scale factor greater than 1 is an enlargement. Review your notes and hapter 6 by using the hapter Review on pages of your textbook. opyright Holt McDougal. ll rights reserved. Words to Review Geometry Notetaking Guide 17