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5-15-1Midsegments of Triangles1. PlanLesson PreviewGO for HelpWhat You’ll LearnCheck Skills You’ll Need To use properties ofFind the coordinates of the midpoint of each segment.midsegments to solveproblems. . . And WhyTo use indirect measurementto find the length of a lake,as in Example 3Lesson 1-8 and page 165Objectives11. AB with A(-2, 3) and B(4, 1) (1, 2)2. CD with C(0, 5) and D(3, 6) A32 , 112BTo use properties ofmidsegments to solveproblemsExamples3. EF with E(-4, 6) and F(3, 10) A–12, 8B1234. GH with G(7, 10) and H(-5, -8) (1, 1)Finding LengthsIdentifying Parallel SegmentsReal-World ConnectionFind the slope of the line containing each pair of points.5. A(-2, 3) and B(3, 1) –526. C(0, 5) and D(3, 6) 317. E(-4, 6) and F(3, 10) 478. G(7, 10) and H(-5, -8) 32Math BackgroundNew Vocabulary midsegment coordinate proof1Euclid did not use coordinategeometry to prove any theorems.The Triangle MidsegmentTheorem can be proved withoutcoordinate geometry, but theproof requires theoremsconcerning parallelogramsthat are not presented in thistext until Chapter 6.Using Properties of MidsegmentsHands-On Activity: Midsegments of TrianglesDraw, label, and cut out a large scalenetriangle. Do the same with other right,acute, and obtuse triangles. Label thevertices A, B, and C.More Math Background: p. 256CCALesson Planning andResourcesBC For each triangle fold A onto C to findthe midpoint of AC . Do the same for BC .Label the midpoints L and N, then draw LN.ALNSee p. 256E for a list of theresources that support this lesson.BPowerPointBell Ringer Practice Fold each triangle on LN.A12 AB;1. LN Explanations may vary. Fold A to C. Fold B to C.2. Answers may vary.Sample: Themidsegment is n tothe 3rd side of the kand is half its length.BCheck Skills You’ll NeedFor intervention, direct students to:B1. How does LN compare to AB?Explain.Finding the Midpoint ofa Segment2. Make a conjecture about how the segment joining the midpoints oftwo sides of a triangle is related to the third side of the triangle.See left.Lesson 1-8: Example 3Extra Skills, Word Problems, ProofPractice, Ch. 1SlopeIn #ABC above, LN is a triangle midsegment. A midsegment of a triangle is asegment connecting the midpoints of two sides.Lesson 5-1 Midsegments of TrianglesSpecial NeedsBelow LevelL1Algebra Review, p. 165259L2In the Hands-On Activity, some students may not seewhy folding A onto C marks the midpoint of ACbecause of its orientation. Demonstrate by folding thecorners of a rectangular piece of paper.To eliminate the fractions in proving Theorem 5-1, letthe respective coordinates of points Q and P be (2a, 0)and (2b, 2c).learning style: visuallearning style: verbal259

2. TeachKey ConceptsTheorem 5-1Triangle Midsegment TheoremIf a segment joins the midpoints of two sides of a triangle, then the segment isparallel to the third side, and is half its length.Guided InstructionHands-On ActivityOne way to prove the Triangle Midsegment Theorem is to use coordinate geometryand algebra. This style of proof is called a coordinate proof. You begin the proofby placing a triangle in a convenient spot on the coordinate plane. You then choosevariables for the coordinates of the vertices.Have students place labels forthe vertices inside the triangleon both sides of the paper so theywill appear on the cut-out figures.Instruct students to label theobtuse or right angle vertex Cto ensure that the first foldedtriangle lies inside ABC.Given: R is the midpoint of OP.S is the midpoint of QP.Connection to AlgebraThe proof of the TriangleMidsegment Theorem uses theMidpoint and Distance Formulasfrom Chapter 1 and the calculationof slope from Chapter 3. Ask:Why are variables used in theproof instead of numbers? Usingnumbers proves the theorem forone set of points. Because anynumber can be substituted fora variable, using variables provesthe theorem for all sets of points.Proof of Theorem 5-1ProofProve: RS 6 OQ and RS 12 OQ Use the Midpoint Formula to find thecoordinates of R and S.Vocabulary TipThe Midpoint Formulax 1 x y 1 yQ 1 2 2, 1 2 2 RThe Distance FormulaSxQ(a, 0)O(0, 0)b 01ca1b cS: Q a 12 , 2 R Q 2 , 2R22 (x2 2 x1) 1 ( y2 2 y1) To prove that RS and OQ are parallel, show that their slopes are equal. Becausethe y-coordinates of R and S are the same, the slope of RS is zero. The same istrue for OQ. Therefore, RS 6 OQ. Use the Distance Formula to find RS and OQ.bb 2cc 2RS % Q a 12 2 2 R 1 Q2 2 2 R22 % Q a2 1 b2 2 b2 R 1 02 % Q a2 R 2a 12 aDiscuss as a class why the verticesin the proof of the TriangleMidsegment Theorem are labeledO(0, 0), Q(a, 0), and P(b, c). Explainthat translating, rotating, orreflecting a triangle so that two ofits vertices are at (0, 0) and (a, 0)simplifies using the Midpoint andDistance Formulas.EXAMPLEP (b, c)Rb 01cb cR: Q 0 12 , 2 R Q 2, 2 RMath Tip1yOQ (a 2 0) 2 1 (0 2 0) 2 a2 1 02 aTherefore, RS 12 OQ.1EXAMPLEFinding LengthsIn #EFG, H, J, and K are midpoints.Find HJ, JK, and FG.Auditory LearnersHJ 21 EG or 12(100); HJ 50Have students read throughExample 1 in small groups. Thenask volunteers to explain howthe example applies the TriangleMidsegment Theorem.JK 12 EF or 12(60); JK 30F60JH40EHK or 40 21 FG; FG 80Quick CheckA1 AB 10 and CD 18. Find EB, BC, and AC.EB 9; BC 10; AC 20EBD260CChapter 5 Relationships Within TrianglesAdvanced LearnersEnglish Language Learners ELLL4Have students use Theorem 5-1 to prove that themidpoints of three sides of a triangle can be used toform four congruent triangles.260GK100learning style: verbalHelp students relate Example 3 with the TriangleMidsegment Theorem. Ask questions such as: Why didDean not just measure the distance across the lake?Why did he mark 35 paces on one side and 118 paceson the other side?learning style: verbal

Connection to Astronomy2EXAMPLEIdentifying Parallel SegmentsIn #DEF, A, B, and C are midpoints. Name pairsof parallel segments.EBAThe midsegments are AB, BC, and CA.By the Triangle Midsegment Theorem,AB 6 DF, BC 6 ED, and AC 6 EFQuick CheckAstronomers use indirectmeasurement to measure greatdistances. Have students researchhow astronomers measuredistances in the universe.DPowerPointFCAdditional ExamplesX2 Critical Thinking Find m&VUZ. Justify your answer.65; UV n XY so lVUZ and lYXZ are corr. and O.65 You can use the Triangle Midsegment Theoremto find lengths of segments that might be difficultto measure directly.UYZV1 In XYZ, M, N, and P aremidpoints. The perimeter of MNP is 60. Find NP and YZ.XM3EXAMPLEReal-WorldConnectionP2224Indirect Measurement Dean plans to swim the length of the lake, as shown in thephoto. How far would Dean swim?Here is what Dean does to find the distance he would swim across the lake.YNZNP 14; YZ 442 Find m&AMN and m&ANM.AStep 1: He measures his stride and adjusts it so that it averages about 3 ft.Step 2: Then he begins at the left edge of the lake (first diagram). He paces35 strides along the edge of the lake and sets a stake.NStep 3: He paces 35 more strides inthe same direction and setsanother stake.35M23635CStep 4: He paces to where his swimwill end at the other side ofthe lake, counting 236 strides.?Step 5: Then (second diagram) he paces118 strides, or half the distance,back towards the second stake.BmlAMN mlANM 753 Explain why Dean could usethe Triangle Midsegment Theoremto measure the length of the lake.He paced between the midpointsof two sides of a triangle.118128Step 6: He paces to the first stake,counting 128 strides.75 ?Resources Daily Notetaking Guide 5-1 L3 Daily Notetaking Guide 5-1—L1Adapted InstructionStep 7: He converts strides to feet.3 ft 384 ft128 strides 3 1 strideStep 8: He uses Theorem 5-1. The distance across the lake is twice the lengthof the midsegment.2(384 ft) 768 ftClosureDean would swim approximately 768 ft.Quick Check3 a. CD is a new bridge being built over a lakeas shown. Find the length of the bridge. 1320 ftb. How long is the bridge in miles? 14 miThe perimeter of a triangle is78 ft. Find the perimeterof the triangle formed by itsmidsegments. 39 ft963 ftC2640 ftBridge963 ftDLesson 5-1 Midsegments of Triangles261261

EXERCISES3. PracticeFor more exercises, see Extra Skill, Word Problem, and Proof Practice.Practice and Problem SolvingAssignment GuideA1 A B 1-36C ChallengePractice by ExampleExample 137-39Test PrepMixed ReviewGO forHelp40-4647-55Mental Math Find the value of x.1. 92.(page 260)3. 147845xx3x7018Homework Quick Check5. 114.To check students’ understandingof key skills and concepts, go overExercises 3, 20, 21, 26, 30.x -145x 16. 25x - 24523 12Error Prevention!Exercise 13 Students may misapplythe Triangle Midsegment Theorem,thinking that the angles are alsoin a 1 : 2 ratio. Review thetheorem with the class beforebeginning this exercise.Points E, D, and H are midpoints of kTUV.UV 80, TV 100, and HD 80.7. Find HE. 409. Find TU. 160Example 2Exercise 29 After students solve(page 261)the exercise, show them how theycan directly find the answer bysimply adding the measures of thediagonals. This follows becauseeach side of the ribbon is half of adiagonal.TEH8. Find ED. 5010. Find TE. 80VUDIdentify pairs of parallel segments in each diagram.11. UW n TX ; UY n VX ;YW n TVUTYVGJ n FK ; JL n HF ; GL n HKH65JG6512.WXF7L713. a. In the figure at the right, identifypairs of parallel segments. ST n PR; SU n QR; UT n PQb. If m&QST 40, find m&QPR.SmlQPR 40PKQTRUName the segment that is parallel to the given segment.GPS Guided Problem Midsegments of Triangles1. In MNO, the points C, D, and E are midpoints. CD 4 cm,CE 8 cm, and DE 7 cm.a. Find MO.b. Find NO.c. Find MN.Example 3(page 261)NCDMOE2. In quadrilateral WVUT, the points F, E, D, and C are midpoints.WU 45 in. and TV 31 in.a. Find CD.b. Find CF.c. Find ED.VECWUDF3. In LOB, the points A, R, and T are midpoints. LB 19 cm,LO 35 cm, and OB 29 cm.a. Find RT.b. Find AT.c. Find AR.20a. 1050 ftTORLTABFind the value of the variable.5.6.7yx2t Pearson Education, Inc. All rights reserved.7. Perimeter of ABC 32 cm8.nB1–n233t21377–n8CL10. QR is a midsegment of LMN.a. QR 9. Find NM.b. LN 12 and LM 31. Find the perimeter of LMN.12.BNY7HGP74I6QCXRRQUse the given measures to identify three pairs of parallel segments ineach diagram.26219. FG CBZGF20. Indirect Measurement Kate wants to paddleher canoe across the lake. To determine howfar she must paddle, she paced out a triangle,counting the number of strides, as shown.a. If Kate’s strides average 3.5 ft, what is thelength of the longest side of the triangle?b. What distance must Kate paddle acrossthe lake? 437.5 ft9.qA618. GE AC4134417. CA EGCUse the diagrams at the right to complete the exercises.A16. EF ABABEL3DatePractice 5-111.15. BC FGL1Adapted Practice4.14. AB FEM262Chapter 5 Relationships Within Triangles8015015080250