Ratio and Proportion. Ratio of a to b If a and b are two quantities that are measured in the same units, then the ratio of a to b is b
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1 8.1 atio and roportion Goals p Find and simplify the ratio of two numbers. p Use proportions to solve real-life problems. VOCABUAY atio of a to b If a and b are two quantities that are measured in the same units, then the ratio of a to b is b a. roportion An equation that equates two ratios is a proportion. Etremes In the proportion b a d c, a and d are the etremes. eans In the proportion b a d c, b and c are the means. Eample 1 implifying atios implify the ratio. 16 kg 7 ft a. b g 2 1 yd To simplify ratios with unlike units, convert to like units so that the units divide out. Then simplify the fraction, if possible. 16 kg 16 p 1000 g 16,000 a g 800 g ft 7 ft 7 b. 2 1 yd 21 p 3 ft Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 164
2 Eample 2 Using atios The perimeter of rectangle JK is 56 centimeters. The ratio of JK : K is 4 : 3. Find the length and width of the rectangle. J l K w Because the ratio of JK : K is 4 : 3, you can represent the length of JK as 4 and the width of K as 3. 2l 2w Formula for perimeter of rectangle 2( 4 ) 2( 3 ) 56 ubstitute for l, w, and ultiply Combine like terms. 4 Divide each side by 14. Answer o, JK has a length of 16 centimeters and a width of centimeters. Checkpoint implify the ratio. 5 km oz 00 m 5 lb 25 : 1 1 : 2 OETIE OF OOTIO 1. Cross roduct roperty The product of the etremes equals the product of the means. If a b c,then ad bc. d 2. eciprocal roperty If two ratios are equal, then their reciprocals are also equal. If a b c b,then d a d c. Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 165
3 Eample 3 olving roportions olve the proportion t 4 t. 8 3 t 4 t 8 3 Write original proportion. 3 (t 4) 8 t Cross product property 3 t 8 t Distributive property 1 2 t implify. 5 Eample 4 olving a roportion A scale model of a car is 10 inches long and 5 inches tall. The actual car is 60 inches tall. What is the length of the actual car? Verbal odel abels easoning ength ofcar Height ofcar e ngth of model He ight of model ength of car Height of car 60 (in.) ength of model 10 Height of model 5 (in.) ubstitute ultiply each side by implify. Answer o, the actual car is 0 inches, or 10 feet long. Checkpoint olve the proportion r 6 8 Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 166
4 8.2 roblem olving in Geometry with roportions Goals p Use properties of proportions. p Use proportions to solve real-life problems. VOCABUAY Geometric mean The geometric mean of two positive numbers a and b is the positive number such that a or a. p b, b ADDITIOA OETIE OF OOTIO 3. If a b c a,then d c b d. 4. If a b c a b c d,then. d b d Eample 1 Using roperties of roportions Tell whether the statement is true. v w If 5,then v 5 w 5. 5 v v w Given w a c a b c d If d,then. b b d w Because w 5, the conclusions are not equivalent. Answer The statement is false. Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 167
5 Eample 2 Using roperties of roportions In the diagram H J G K. Find the length of &*. GK J K H J G K J K Given. H 6 J 6 ubstitute G K Cross product property 5 Divide each side by. Answer o, the length of &* GK is 5. Eample 3 Using a Geometric ean The two buildings shown have the same width-to-length ratio. The distance labeled is the geometric mean of 105 m and 210 m. Find the value of. 105 m 210 m Write proportion p p p2 p Cross product property implify. Factor implify. Answer The geometric mean of 105 and 210 is 105 2, or about 148. o, the distance labeled in the diagram is about 148 m. Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 168
6 Checkpoint Complete the following eercises. 1. Tell whether the statement is true: If m 7 n 11,then m n True 2. In the diagram, T W X. Find T. TV VW T 8 V 21 W X 3. Find the geometric mean of 54 and. 722 Eample 4 olving a roportion You have a 10-by-8 inch photo of the school band that must be reduced to a length of 5.5 inches for the school yearbook. What is the width of the reduced photo? 8 in. 10 in. 5.5 in. Verbal odel ength of original ength of reduced Width of original Width of reduced abels ength of original photo 10 ength of reduced photo 5.5 Width of original photo 8 Width of reduced photo (inches) (inches) (inches) (inches) easoning ubstitute ( 5.5 ) Cross product property 4.4 Use a calculator. Answer o, the reduced photo has a width of 4.4 inches. Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 16
7 8.3 imilar olygons Goals p Identify similar polygons. p Use similar polygons to solve real-life problems. VOCABUAY imilar polygons imilar polygons are two polygons such that their corresponding angles are congruent and the lengths of corresponding sides are proportional. cale factor The scale factor is the ratio of the lengths of two corresponding sides of two similar polygons. Eample 1 Comparing imilar olygons Decide whether the figures are similar. If they are similar, write a similarity statement. J F 8 G K H As shown, the corresponding angles of FGHJ and K are congruent. Also, the corresponding side lengths are proportional. F G 8 K G H HJ FJ 1 2 K 4 3 Answer o, the two figures are similar and you can write FGHJ K. Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 170
8 Checkpoint Decide whether the figures are similar. If they are, write the similarity statement. 1. A 8 B D 7 E C F The triangles are not similar. Eample 2 Comparing hotographic Enlargements You have a 4-inch by 6-inch photo that you want to use for class election posters. You want the enlargement to be 26 inches wide. How long will it be? 4 6 Compare the enlargement to the original measurements of the photo. 2 6 in. in. 4 in. 6 in p inches Answer The length of the enlargement will be 3 inches. THEOE 8.1 If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths. If K Q, then K K Q Q K Q K K. Q Q Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 171
9 Eample 3 Using imilar olygons entagon CDFGH is similar to pentagon JK. Find the value of. et up a proportion that contains CD. H G C D F J 2 K 6 C D JK DF K Write proportion. 2 6 ubstitute. 3 Cross multiply and divide by 6. Checkpoint Complete the following eercises. 2. Verify that these two triangles are similar. Write the similarity statement. Then find the ratio of their perimeters. These triangles are similar because the corresponding angles are congruent and the corresponding sides are in the ratio 3 : 2; Q VT; 3 : 2 Q V 6 T 3. arallelogram JK is similar to parallelogram Q. Find the value of J 20 8 K Q Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 172
10 8.4 imilar Triangles Goals p Identify similar triangles. p Use similar triangles in real-life problems. Eample 1 Writing roportionality tatements In the diagram, TGT TG. a. Write the statement of proportionality. b. Find mag. c. Find G G G a. G GT T G 20 T b. at cag, so mag 38. c. G T GT Write proportion. 15 G ubstitute. 20 (15) 24 G ultiply each side by G implify. Answer o, G is.5 units. OTUATE 25: AGE-AGE (AA) IIAITY OTUATE If two angles of one triangle are K congruent to two angles of another Y triangle, then the two triangles are similar. If ajk caxyz and akj cayxz, then T JK T XYZ. J X Z Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 173
11 Eample 2 roving that Two Triangles are imilar In the diagram, TABH TKH. Use properties of similar triangles to eplain why these triangles are similar. A You can use the Vertical Angles Theorem to determine aahb cakh. Because they are right angles, aabh cakh. By the AA imilarity ostulate, you can conclude that TABH TKH. B H K Eample 3 Using imilar Triangles To comply with the Americans with Disabilities Act, wheelchair ramps made for new constructions must have a height to length ratio of 1 :. At a new construction, the height h to the bottom of a door is 2.5 feet. Use the proportion a b h to estimate the length r r that the ramp should be to have the correct slope ratio. In the proportion, use a 1 ft and b ft. h a b r a b h Write proportion. r 1 ft ft 2.5 ft r ubstitute. r 30 ft Cross product property Answer The ramp should have a length of 30 feet. Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 174
12 Eample 4 Using cale Factors Find the length of &*. First, find the scale factor of TDF to TT. D 15 G 15 D T 18 3 F T ow, because the ratio of the lengths of the segments is equal to the scale factor, you can write the following equation. FG 5 3 Answer ubstitute for FG and solve for to show that 7.2. Checkpoint Complete the following eercises. 1. You are standing 15 m from 2. TJ TT. Find the building A and 50 m from building B. Building A is length of K &*. 0 m tall. Find the height of J building B. 16 B A 0 m K T 50 m 15 m 300 m 10.5 Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 175
13 8.5 roving Triangles are imilar Goals p Use similarity theorems to prove two triangles are similar. p Use similar triangles to solve real-life problems. THEOE 8.2: IDE-IDE-IDE () IIAITY THEOE If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar. A Q AB BC CA If Q, B C Q then T ABC TQ. THEOE 8.3: IDE-AGE-IDE (A) IIAITY THEOE If an angle of one triangle is congruent X to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. Z ZX XY If ax ca and, then T XYZ T. Y Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 176
14 Eample 1 Using the imilarity Theorem Which of the following three triangles are similar? Q W 6 V 15 To decide which, if any, of the triangles are similar, you need to consider the ratios of the lengths of corresponding sides. atios of ide engths of TQ and TTVW hortest sides ongest sides emaining sides 4 V W 6 2 3, Q 1 0 TW , Q 8 2 TV Answer Because the ratios are equal, TQ TTVW. atios of ide engths of TQ and TXYZ hortest sides ongest sides emaining sides 4 YZ 6 2 3, Q 1 0 XZ 5 6, Q 8 4 XY Answer Because the ratios are not equal, TQ and TXYZ are not similar. T Z Y 6 10 X Checkpoint Complete the following eercise. 1. Which of the three triangles are similar? TK and TQ K Q 15 G 14 J H Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 177
15 Eample 2 Using the A imilarity Theorem Use the given lengths to prove that TDF T. D 2 10 Given: DF 15, D 2, D rove: TDF T aragraph roof Use the A imilarity Theorem. Begin by finding the ratios of the lengths of the corresponding sides. F DF D o, the lengths of sides DF *& and D *& are proportional to the lengths of the corresponding sides of T. Because afd and a are right angles, use the A imilarity Theorem to conclude that TDF T. Checkpoint Complete the following eercise. 2. Describe how to prove that TV is similar to TYXV X how that corresponding sides are proportional. Then use the A imilarity Theorem with the vertical angles av and ayvx. 20 V 18 Y Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 178
16 8.6 roportions and imilar Triangles Goals p Use proportionality theorems to calculate segment lengths. p olve real-life problems. THEOE 8.4: TIAGE OOTIOAITY THEOE If a line parallel to one side of a triangle Q T intersects the other two sides, then it divides the two sides proportionally. T If TU& Q &*,then T Q U. U U THEOE 8.5: COVEE OF THE TIAGE OOTIOAITY THEOE If a line divides two sides of a triangle Q proportionally, then it is parallel to the third side. T If U, then TU *& Q *&. T Q U T U Eample 1 Finding the ength of a egment What is the length of &*? Q Theorem ubstitute. 6 4 Q 4( ) ultiply each side by. 6 6 implify. Answer o, the length of &* is 6. Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 17
17 Checkpoint Complete the following eercises. 1. Find the length of DF &*. D F 6 H G 8 J 2. Given the diagram, determine whether &* is parallel to Q& Q 48 no THEOE 8.6 If three parallel lines intersect two transversals, then they divide the transversals proportionally. r s t l U W Y V X Z m If r s and s t, and l and m intersect r, s, and t, then U W V X. WY XZ THEOE 8.7 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. C A D B If CD wy bisects aacb, then A D DB C A. CB Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 180
18 Eample 2 Using roportionality Theorems What is the length of GH &*? Because corresponding angles are congruent, the lines are parallel and you can use Theorem 8.6. F 28 J 30 G K 25 H FG JK G H K arallel lines divide transversals proportionally ubstitute. GH p p GH Cross product property. 7 0 GH Divide each side by 30 and simplify. 3 Answer o, the length of GH &* is 7 0,or Eample 3 Using roportionality Theorems In the diagram, aq ca. Use the given side lengths to find the length of Q &*. Because &* is an angle bisector of aq, you can apply Theorem 8.7. et Q. Then, 18. Apply Theorem 8.7. Q Q Q ubstitute. 8 p ( 18 ) Cross product property Distributive property Add to each side Divide each side by 20. Answer o, the length of Q &* is 10.8 units. Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 181
19 Eample 4 Finding the ength of a egment Hiking Trails A state park has five trails, as shown in the diagram. The horizontal ed, Green, and Blue trails are evenly spaced. The iver and ky trails intersect the ed, Green and Blue trails. Eplain why the distance between the ed and Green trails on the ky trail is the same as the distance between the Green and Blue trails on the ky trail. ed A B Green C D Blue E F iver Because the ed, Green, and Blue trails are each horizontal, you know that they are parallel to each other. Using Theorem 8.6, you can conclude that A C B D. Because the trails are evenly spaced, CE DF you know that AC CE. o, you can conclude that BD DF, which means that the distance between the ed and Green trails and the Green and Blue trails on the ky trail have the same lengths. ky Checkpoint Complete the following eercises. 3. In the diagram, HK &* &* &*, H 2.4, K 2.8, and 4.. Find the length of &*. H K Find the value of Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 182
20 8.7 Dilations Goals p Identify dilations. p Use properties of dilations in real-life. VOCABUAY Dilation A dilation with center C and scale factor k is a transformation that maps every point in the plane to a point so that the following properties are true. (1) If is not the center point C, then the image point lies on C &*(. The scale factor k is a positive number such that k C, and k 1. (2) If is the center point C, then C. Enlargement A dilation with k > 1 eduction A dilation with 0 < k < 1 Eample 1 Identifying Dilations Identify the dilation and find its scale factor. a. b. 5 3 C 5 C a. Because C C 3 5,the scale factor is k 3 5. This is a reduction. b. Because C C 1 2,the scale factor is k 5. This is 5 an enlargement. Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 183
21 Eample 2 Dilation in a Coordinate lane Draw a dilation of TXYZ. Use the origin as the center and use a scale factor of 2. Find the perimeter of the preimage and the perimeter of the image. Because the center of the dilation is the origin, you can find the image of 7 X each verte by multiplying its 5 X coordinates by the scale factor. 3 X(1, 4) X( 2, 8 ) Y 1 Y(1, 1) Y( 2, 2 ) Y Z Z Z(5, 1) Z( 10, 2 ) To find the perimeters of the preimage and image, you need to find XZ and XZ. XZ ( 1 5 ) 2 ( 4) 1 2 ( ) erimeter of TXYZ XZ ( 2 10 ) 2 ( 8 2) 2 ( 8 ) erimeter of TXYZ y 2 Checkpoint Use the dilation shown. 1. Is the dilation shown a reduction or an enlargement? 5 y eduction 2. What is the scale factor? Q 3 7 Q 3. What are the coordinates of the vertices of the image? 5 (3, 2), (4, 0), (3, 2), Q(2, 0) Copyright cdougal ittell/houghton ifflin Company All rights reserved. Chapter 8 Geometry otetaking Guide 184
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