For use with pages 357-3B3 ratio is a comparison of a number a and a nonzero number b using division. n equation that states that two ratios are equal is called a proportion. In the proportion a ~ = c ~, the numbers b and c are the means of the proportion. The numbers a and d are the extremes of the proportion. Cross Product Property In a proportion, the product of the extremes is equal to the product of the means. Simplify the ratio. a. 6 days:15 days b. 2ft 2 yd a, 6 days:15 days can be written as the fraction 6 days 15 days" 6 days 6 + 3 Divide numerator and denominator by theft 15 days 15 + 3 greatest common factor, 3. b= 2ft 2yd 2ff 2.3ft 2 6 1 2 Simplify. 2 ~ is read "2 5? as to Substitute 3 ft for 1 yd. Multiply. Divide numerator and denominator by 1 their greatest common factor, 2. ~ is read "1 to 3." Si~pJify the ratio ~. 6 in.:28 in. 2. 18 6cm cm 3. 27 3ft in. Copyright McDougal Littell inc.
For use with pages 357-353 Use Ratios In the diagram XY: YZ is 1!5 and XZ = 24. [ Find XY and YZ. X Y 24 Let x = XY. Because the ratio of XY to YZ is 1 to 5, you know that YZ = 5x. XY + YZ = XZ Segment ddition Postulate x + 5x = 24 Substitute x for XY, 5x for YZ, and 24 for XZ. 6x = 24 dd like terms. x = 4 Divide each side by 6. To find XY and YZ, substitute 4 for x. XY- x = 4 YZ = 5x = 5 4 = 20 nswer: So, XY = 4 and YZ = 20. X x y 5x Z Find the segment ~engthso I 4. In the diagram, B:BC is 2:1 and C = 15. ~.~ Find B and BC. 5. In the diagram, DE:EF is 4:9 and DF = 39. Find DE and EF. In the diagram, JK:KL is 6:7 and JL = 26. Find JK and KL. 15 39 26 3 9 Solve the proportion -~ - x - 1" 3 9 2 x-1 3(x-- 1)=2"9 3x-- 3 = 18 3x = 21 x=7 Write original proportion. Cross Product Property Multiply and use the Distributive Property. dd 3 to each side. Divide each side by 3. Se~ve the proportien. 7. x 7 2-- 14 Copyright McDougal Littell Inc. 8. ~--- 5_y+1 21 9, 27 3 x-5 2
Two polygons are similar polygons if corresponding angles are congruent and corresponding side lengths are proportional. If the two polygons are similar, then the ratio of the lengths of two corresponding sides is called the scale factor. Theorem 7.1 Perimeters of Similar Polygons If two polygons are similar, then the ratio of their perimeters is equal to the ratio of their corresponding side lengths. Use $imilarit~ Statements JKLM,--- WXYZ a. List all pairs of congruent angles. b. Write the ratios of the corresponding sides in a statement of proportionality.. Check that the ratios of corresponding sides are equal. WX XY YZ WZ JK KL LM JM WX 3 XY 12 3 YZ 6 3 WZ 9 ~"... JK 4 KL 16 4 LM 8-4 and JM 12-3 The ratios of corresponding sides are all equal to ~. 3 1. /klmn ~ /k RST M H 12 2. BCD ~ EFGH 18 N 16 L B4~C 6 6 12V D 6 R 8 Copyright McD0ugal Littell Inc.
For use with pages 364-371 Determine whether the quadrilaterals are similar. C If they are similar, write a similarity statement and find the scale factor of FGHE to BCD. B D F From the diagram, you know that the corresponding angles are congruent because / ~ Z F, /B ~- / G, Z C = Z H, and Z D ----- Z E. The corresponding side lengths are proportional because the following ratios are equal. FG 5 GH 10 5 HE 15 5 FE 20 5 B 7 BC 14 7 CD 21 7 D 28 7 The quadrilaterals are similar. FGHE--BCD. Thescale factor of FGHE to BCD 5 is 7" Exercise for Example 2 3. Determine whether the polygons are similar. If they are similar, write a similarity statement and find the scale factor of Figure B to Figure. 10V10 S 15 15 Y R T In the diagram, JKLM--~PQRS. Find the value of x. Because the quadrilaterals are similar, the corresponding side lengths are proportional. To find the value of x, you can use the following proportion. RS _ PQ LM JK Write proportion. 7-21 3 x Substitute 7 for RS, 3 for LM, 21 for PQ, and x for JK. 7x = 63 Cross Product Property x = 9 Divide each side by 7. 21 the diagram,/k BC ~ /k XYZ. Find the va~ue of x. 4. B 5. Z 15 3 x 10 8 C Copyright McDougal Littell Inc. Y X Chapter 7,Resource Book
For.ss wi~h pages 372-378 I P ostulace 15 ngle-ngle () S~rnflar~ty PosCullate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning. Y If two pairs of angles are congruent, then the triangles are similar. Z ~- ZZ because they both have a measure of 52. Z C is not congruent to ZX because their measures are not equal (88 =# 40 ). So find mz Y to see whether Z C is congruent to Z K mz Y + 52 + 40 = 180 Triangle Sum Theorem mz Y + 92 = 180 dd. mz Y = 88 Subtract 92 from each side. Both Z C and Z Y measure 88, so Z C --- Z Y. By the Similarity Postulate,. /~ B C - ZXY. Determine whether the triangles are similar, ff they are similar, write a similarity statement. ~. B F 2. H G / C 3. N R 4. M 83 P Q Copyright McD0ugal Littell Inc.
For use with pages 372-378 Given that E bisects /BD, is there enough information to show that/~ BC is similar to x DE? Explain your reasoning. From the diagram, you know that Z B ~ Z D. By the definition of an angle bisector, you know that Z BC ~ Z DE. So,/~ BC --~,V DE by the Similarity Postulate. Exemises for Example 2,, 78 D J N M T In the diagram, P =~ I~ and ~ [S ~. Find the value of x. 16 F 24 R Corresponding angles of parallel lines are,congruent, so Z QPR ~ Z SRT and Z QRP ~- Z STR. By the Similarity Postulate, PQR--~ RST. To find the value of x, set up the following proportion. RS RT p-~ = fi~- Write a proportion. 10 x 16 24 Substitute 10 for RS, 16 for PQ, x for RT, and 24 for PR. 240 = 16x Cross Product Property 15=x Divide each side by 16. T Write a similaritv statement fer the triangles. Then find the va~ue of the variame. 20 X 15~ 12 Geo~:~f p 12 D, Copyright McD0ugal Littell Inc. ll rights resewed.
For use with pages 379-385 Show that two triangles are similar usincj the SSS and SS Similarity Theorems. i Theorem 7.2 SideoSideoSide (SSS) S~m~lar~y Theorem ~ If the co~esponding sides of two triangles ~e propo~ional, then the triangles ~e si~l~. ~ Theorem 7.3 Side-ngle-Side (SS) Similarity Theorem If an angle of one ~angle is congruent to an angle of a second triangle and the lengths of the sides that include these angles ~e proportional, then the triangles ~e si~l~. Determine whether the triangles are similar. If they are similar, write a similarity statement and find the scale factor of/~ DEF to/~ BC. Find the ratios of the corresponding sides. DE 14 + 2 7 B 18 + 2 9 EF 21 21 + 3 7 -- BC 27 27 + 3 9 DF 28 28 + 4 7 C 36 36 + 4 9 ll three ratios are equal. So, the corresponding.sides of the triangles are proportional. By the SSS Similarity Theorem,/~ BC--/~ DEF. The scale factor 7 of/~ DEF to/~ BC is ~. Exercises for Example I Determine whether the triangles are similar. ~f they are similar, write a similarity statement and find the sca~e factor of triangle B to triangle. 1. K N 2. R 27 38 27 44 49 2 30 30 J 32 L B S M 36 P Q 40 X CopYright McD0ugal Littell Inc. ll rights reserved
For use with pages 379-385 Determine whether the triangles are similar. R If they are similar, write a similarity statement. K 55 22 /L and Z R both measure 79, so Z L ----- /R. 0 Compare the ratios of the side lengths that include /L and Z R. Q ~_R _ 55 _ 55 11 Shorter sides: RS 22 _ 22 + 2 _ 11 Longer Sides: ML 25 25 + 5 - ~ LK 10 10 + 2 5 The lengths of the sides that include Z L and Z R are proportional. By the SS Similarity Theorem,/~ KLM --- /~ SRQ. 3. B D 7 E T 12 14 c S O 10 20 Show that/~ CD,~ /~ BE. C Separate/~ CD and/~ BE and label the side lengths. B C D 12 E 9 12 D 21 Z ~ Z by Reflexive Property of Congruence. Shorter sides: --C-C = 8 +(5 =.14 _ 7 Longer sides: --D-D -- 12 + 9 _, 21 _ B 8 8 4 E 12 12 The lengths of the sides that include Z are proportional. By the S/kS Similarity Theorem,/~ CD --~ /k BE. Show that the overlapping triangles ~e si~l~. D 7 E 7 F Then write a similarity statement. Copyright McDougal Littell Inc.
For use with pages Use the Triangle Propo~tienMity Theorem and its converse. midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle. Theorem 7.4 Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. Theorem 7.5 Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side. Theorem 7.6 The Midsegment Theorem The segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long.! Find the value of x. R P x T 12 S PQ _ PT QR TS 32 x 16 12 32. 12= 16.x 384 = 16x 384 16x 16 16 24 = x Triangle Proportionality Theorem Substitute 32 for PQ, 16 for QR, x for PT, and 12 for TS. Cross Product Property Multiply. Divide each side by 16. Simplify. F~nd the vmue ~ the ~, ~x B 10 C 2.,~ ~ ~" ~ L ~~ N E J Copyright @ McOougal Littell Inc.
For use with pages 388-392 Determine Pemfie Given the diagram, determine whether B~" is parallel to ~-~. ~ E Find a._..nd simplify the ratios of the two sides divided by BE. B 9 3 E 12 3 C BC 6 2 ED 8 2 The ratios are equal, so the two sides divided by BE are proportional. By the Converse of the Triangle Proportionality Theorem, BE is parallel to CD. 3. 4. C C D D 28 Find the value of the variable. It is given in the diagram that KL = JK = 8, so K is the midpoint of ~. It is also given in the diagram that JN = MN = 10, so N is the midpoint of JM. Therefore, KN is a midsegment of/~ JLM. Use the Midsegment Theorem to write the following equation. KN = -~ LM The Midsegment Theorem 1 13 = ~x Substitute 13 for KN and x for LM. x- 26 Multiply each side by 2. Exercises for Example 3 Find the va~ue e[ the var~ame. 5. 7 J 10 N 10 M 42 Copyright McDougal Littell Inc.
dilation is a transformation with center C and scale factor k that maps each point P to an image point P so that P lies on CP and CP = k. CP. dilation is called a reduction if the image is smaller than the original figure. dilation is called an enlargement if the image is larger than the original figure. Tell whether the dilation is a reduction or an enlargement. a. B b. E B! E! E D C a. The dilation is a reduction because the image ( B D E ) is smaller than the original figure (BDE). b, The dilation is an enlargement because the image ( E F G ) is larger than the original figure ( EFG)...... Tel~,whether the dilation is a reduction or an emargement. = R I S I p, P C 3. X Y y! W z Z! Copyright McDougal Littell Inc.
For use with pages 393-399 Find the scale factor of the dilation. Find the ratio of CP to CP. scale factor CP _ 21 _ 3 b. scale factor = CP [ _ 22 _ 11, CP 14 2 CP 40 20 4, I 32 I 14 p /~ P D is the image of PD after an enlargement. ~--- 24 ---1 Find the value of x. C 9 P P CP P D CP PD 24 x 9 15 :360 = 9x., 40 = x Write a proportion. Substitute 24 for CP, 9 for CP, x for P D, and 15 for PD. Cross Product Property Divide each side by 9. O! Copyright McDougal Littell Inc.