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8.4Proportionality TheoremsEssential QuestionWhat proportionality relationships exist in atriangle intersected by an angle bisector or by a line parallel to one of the sides?Discovering a Proportionality RelationshipWork with a partner. Use dynamic geometry software to draw any ABC.— parallel to BC— with endpoints on AB— and AC—, respectively.a. Construct DEBDCEALOOKINGFOR STRUCTURETo be proficient in math,you need to look closelyto discern a patternor structure.b. Compare the ratios of AD to BD and AE to CE.— to other locations parallel to BC— with endpoints on AB— and AC—,c. Move DEand repeat part (b).d. Change ABC and repeat parts (a)–(c) several times. Write a conjecture thatsummarizes your results.Discovering a Proportionality RelationshipWork with a partner. Use dynamic geometry software to draw any ABC.a. Bisect B and plot point D at theintersection of the angle bisector—.and ACb. Compare the ratios of AD to DCand BA to BC.BADCc. Change ABC and repeat parts (a)and (b) several times. Write aconjecture that summarizesyour results.Communicate Your AnswerB3. What proportionality relationships exist in a triangleintersected by an angle bisector or by a line parallelto one of the sides?4. Use the figure at the right to write a proportion.Section 8.4hs geo pe 0804.indd 445DEAProportionality TheoremsC4451/19/15 12:25 PM
8.4 LessonWhat You Will LearnUse the Triangle Proportionality Theorem and its converse.Use other proportionality theorems.Core VocabulVocabularylarryPreviouscorresponding anglesratioproportionUsing the Triangle Proportionality TheoremTheoremsTheorem 8.6 Triangle Proportionality TheoremQIf a line parallel to one side of a triangleintersects the other two sides, then it dividesthe two sides proportionally.TRUSRTRUIf —TU —QS , then — —.TQUSProof Ex. 27, p. 451Theorem 8.7 Converse of the Triangle Proportionality TheoremQIf a line divides two sides of a triangleproportionally, then it is parallel to thethird side.TRUSRTRUTU —QS .If — —, then —TQUSProof Ex. 28, p. 451Finding the Length of a Segment— UT—, RS 4, ST 6, and QU 9. What is the length of RQ—?In the diagram, QSR4Q9S6UTSOLUTION— —RQQURSSTTriangle Proportionality Theorem— —RQ946Substitute.RQ 6Multiply each side by 9 and simplify.— is 6 units.The length of RQMonitoring Progress—1. Find the length of YZ .Help in English and Spanish at BigIdeasMath.comV35W44YX36Z446Chapter 8hs geo pe 0804.indd 446Similarity1/19/15 12:25 PM
The theorems on the previous page also imply the following:Contrapositive of the TriangleProportionality TheoremRT RU— QS—.If — —, then TUTQ USInverse of the TriangleProportionality TheoremRT RU— QS—, then — —.If TUTQ USSolving a Real-Life ProblemOn the shoe rack shown, BA 33 centimeters,CB 27 centimeters, CD 44 centimeters, andDE 25 centimeters. Explain why the shelf is notparallel to the floor.CBDAESOLUTIONFind and simplify the ratios of the lengths.44CB 279— — —25BA 33 119 —44—. So, the shelf is not parallel to the floor.Because — —, BDis not parallel to AE25 11CDDE— —PQ50Monitoring Progress90N72Help in English and Spanish at BigIdeasMath.com— —2. Determine whether PS QR .S 40 RRecall that you partitioned a directed line segment in the coordinate plane in Section3.5. You can apply the Triangle Proportionality Theorem to construct a point along adirected line segment that partitions the segment in a given ratio.Constructing a Point alonga Directed Line Segment— so that the ratio of AL to LB is 3 to 1.Construct the point L on ABSOLUTIONStep 1Step 2Step 3CCCGGFFEEDABDraw a segment and a ray— of any length. Choose anyDraw ABpoint C not on ⃖ ⃗AB. Draw ⃗AC.ADBDraw arcs Place the point ofa compass at A and make an arcof any radius intersecting ⃗AC . Labelthe point of intersection D. Using thesame compass setting, make threemore arcs on ⃗AC, as shown. Label thepoints of intersection E, F, and G andnote that AD DE EF FG.Section 8.4hs geo pe 0804.indd 447AJKLB—. Copy AGBDraw a segment Draw GBand construct congruent angles at D, E,— at J, K,and F with sides that intersect AB—, EK—, and FL— are alland L. Sides DJ—parallel, and they divide AB equally. So,AJ JK KL LB. Point L dividesdirected line segment AB in the ratio 3 to 1.Proportionality Theorems4471/19/15 12:25 PM
Using Other Proportionality TheoremsTheoremTheorem 8.8 Three Parallel Lines TheoremIf three parallel lines intersect two transversals,then they divide the transversals proportionally.rstUWYVXZUWWYmVXXZ— —Proof Ex. 32, p. 451Using the Three Parallel Lines TheoremIn the diagram, 1, 2, and 3 are all congruent,GF 120 yards, DE 150 yards, andCD 300 yards. Find the distance HF betweenMain Street and South Main Street.1Main St.120 yd2GSOLUTIONE150 ydD Second St.300 ydCorresponding angles are congruent, so⃖ ⃗FE, ⃖ ⃗GD, and ⃖ ⃗HC are parallel. There aredifferent ways you can write a proportionto find HG.Method 1F3HSouth Main St.CUse the Three Parallel Lines Theorem to set up a proportion.— —HGGFCDDEThree Parallel Lines Theorem— —HG120300150Substitute.HG 240Multiply each side by 120 and simplify.By the Segment Addition Postulate (Postulate 1.2),HF HG GF 240 120 360.The distance between Main Street and South Main Street is 360 yards.Method 2Set up a proportion involving total and partial distances.Step 1 Make a table to compare the distances.Total distance⃖ ⃗CE⃖ ⃗HFCE 300 150 450HFDE 150GF 120Partial distanceStep 2 Write and solve a proportion.450150HF120— —Write proportion.360 HFMultiply each side by 120 and simplify.The distance between Main Street and South Main Street is 360 yards.448Chapter 8hs geo pe 0804.indd 448Similarity1/19/15 12:25 PM
TheoremTheorem 8.9 Triangle Angle Bisector TheoremIf a ray bisects an angle of a triangle, thenit divides the opposite side into segmentswhose lengths are proportional to thelengths of the other two sides.ADCBAD CA— —DBCBProof Ex. 35, p. 452Using the Triangle Angle Bisector Theorem—.In the diagram, QPR RPS. Use the given side lengths to find the length of RSQ7PR1315xSSOLUTIONBecause ⃗PR is an angle bisector of QPS, you can apply the Triangle Angle BisectorTheorem. Let RS x. Then RQ 15 x.RQ PQRSPS715 x— —x13195 13x 7x— —Triangle Angle Bisector TheoremSubstitute.Cross Products Property9.75 xSolve for x.— is 9.75 units.The length of RSMonitoring ProgressHelp in English and Spanish at BigIdeasMath.comFind the length of the given line segment.—3. BD—4. JM40A16EM3C2G1B DHF3016 J15K18NFind the value of the variable.S5.24V146.4Tx484 2USection 8.4hs geo pe 0804.indd 449YWZ4yXProportionality Theorems4491/19/15 12:25 PM
Exercises8.4Dynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. COMPLETE THE STATEMENT If a line divides two sides of a triangle proportionally, then it isto the third side. This theorem is known as the .—2. VOCABULARY In ABC, point R lies on BC and ⃗AR bisects CAB. Write the proportionalitystatement for the triangle that is based on the Triangle Angle Bisector Theorem (Theorem 8.9).Monitoring Progress and Modeling with Mathematics—.In Exercises 3 and 4, find the length of AB(See Example 1.)3.AAE 4.14E12B3C4In Exercises 13–16, use the diagram to complete 25LN7.K1510NJECDF16. — —EGCE15. — ———17. VX18. SUY3416M15LCGCEBDIn Exercises 17 and 18, find the length of the indicatedline segment. (See Example 3.)35KBFDFCGCGJ8.24MG14. — —N18B13. — —(See Example 2.)LDC— JN—.In Exercises 5–8, determine whether KM5.FZ20W8U15P 8 R T12XN S10VUIn Exercises 19–22, find the value of the variable.(See Example 4.)19.CONSTRUCTION In Exercises 9–12, draw a segment with8the given length. Construct the point that divides thesegment in the given ratio.20.yz31.54.5469. 3 in.; 1 to 421.10. 2 in.; 2 to 3p16.511. 12 cm; 1 to 31122.16q36282912. 9 cm; 2 to 5450Chapter 8hs geo pe 0804.indd 450Similarity1/19/15 12:25 PM
23. ERROR ANALYSIS Describe and correct the error in29. MODELING WITH MATHEMATICS The real estate termsolving for x. lake frontage refers to the distance along the edge of apiece of property that touches a lake.xADC14lake1016174 ydBAB CD ——BC ADLot A10 14 ——x1610x 224x 22.448 yda. Find the lake frontage (to the nearest tenth) ofeach lot shown.b. In general, the more lake frontage a lot has, thehigher its selling price. Which lot(s) should belisted for the highest price?the student’s reasoning.Bc. Suppose that lot prices are in the same ratio as lakefrontages. If the least expensive lot is 250,000,what are the prices of the other lots? Explainyour reasoning.DALot C61 yd55 ydLakeshore Dr.24. ERROR ANALYSIS Describe and correct the error in Lot BCBD ABBecause — — and BD CD,CD ACit follows that AB AC.30. USING STRUCTURE Use the diagram to find thevalues of x and y.2MATHEMATICAL CONNECTIONS In Exercises 25 and 26,— RS—.find the value of x for which PQP 2x 425.S 5RQP26.T12R2x 2T73x 51.55Qx321yS3x 131. REASONING In the construction on page 447, explainwhy you can apply the Triangle ProportionalityTheorem (Theorem 8.6) in Step 3.27. PROVING A THEOREM Prove the TriangleProportionality Theorem (Theorem 8.6).— TU—Given QSQT SUProve — —TR UR32. PROVING A THEOREM Use the diagram with theQTRUSauxiliary line drawn to write a paragraph proof ofthe Three Parallel Lines Theorem (Theorem 8.8).Given k1 k2 k3Prove28. PROVING A THEOREM Prove the Converse of theCBBADEEF— —Triangle Proportionality Theorem (Theorem 8.7).ZYZXGiven — —YW XVProveCW Y— WV—YXBZVXASection 8.4hs geo pe 0804.indd 451t1Dt2EFauxiliarylinek1k2k3Proportionality Theorems4511/19/15 12:25 PM
33. CRITICAL THINKING In LMN, the angle bisector of—. Classify LMN as specifically as M also bisects LNpossible. Justify your answer.38. MAKING AN ARGUMENT Two people leave points Aand B at the same time. They intend to meet atpoint C at the same time. The person who leavespoint A walks at a speed of 3 miles per hour. You anda friend are trying to determine how fast the personwho leaves point B must walk. Your friend claims you—. Is your friend correct?need to know the length of ACExplain your reasoning.34. HOW DO YOU SEE IT? During a football game,the quarterback throws the ball to the receiver. Thereceiver is between two defensive players, as shown.If Player 1 is closer to the quarterback when the ballis thrown and both defensive players move at thesame speed, which player will reach the receiverfirst? Explain your reasoning.AB0.9 mi0.6 miDCE39. CONSTRUCTION Given segments with lengths r, s,r tand t, construct a segment of length x, such that — —.s xrs35. PROVING A THEOREM Use the diagram with theauxiliary lines drawn to write a paragraph proof ofthe Triangle Angle Bisector Theorem (Theorem 8.9).t40. PROOF Prove Ceva’s Theorem: If P is any point AY CX BZinside ABC, then — — — 1.YC XB ZAGiven YXW WXZYW XYProve — —WZ XZNYXAauxiliary linesBZMXPWZAYC— through A and C,(Hint: Draw segments parallel to BYas shown. Apply the Triangle Proportionality Theorem(Theorem 8.6) to ACM. Show that APN MPC, CXM BXP, and BZP AZN.)36. THOUGHT PROVOKING Write the converse of theTriangle Angle Bisector Theorem (Theorem 8.9).Is the converse true? Justify your answer.37. REASONING How is the Triangle Midsegment Theorem(Theorem 6.8) related to the Triangle ProportionalityTheorem (Theorem 8.6)? Explain your reasoning.Maintaining Mathematical ProficiencyReviewing what you learned in previous grades and lessonsUse the triangle. (Section 5.5)41. Which sides are the legs?a42. Which side is the hypotenuse?cbSolve the equation. (Skills Review Handbook)43. x2 121452Chapter 8hs geo pe 0804.indd 45244. x2 16 2545. 36 x2 85Similarity1/19/15 12:25 PM
_ to the third side. This theorem is known as the _. 2. VOCABULARY In ABC, point R lies on BC — and AR ⃗ bisects CAB. Write the proportionality statement for the triangle that is based on the Triangle Angle Bisector Theorem (Theorem 8.9). VVocabula
