Concurrent Lines, Medians, and Altitudes 273

Home , Altitude (triangle), Concurrent lines, Median (geometry)

5-3 5-3 Concurrent Lines, Medians, 5-3 and Altitudes 1. Plan

Objectives What You’ll Learn Check Skills You’ll Need GO for Help Lesson 1-7 1 To identify properties of • To identify properties of perpendicular bisectors and For Exercises 1–2, draw a large triangle. Construct each ﬁgure. 1–4. See perpendicular bisectors and back of book. angle bisectors angle bisectors 1. an angle bisector 2 To identify properties of • To identify properties of 2. a perpendicular bisector of a side medians and altitudes of a medians and altitudes of a * ) GH. CD ' GH GH. triangle triangle 3. Draw * ) Construct * at the) midpoint* of) * ) Examples . . . And Why 4. Draw AB with a point E not on AB . Construct EF ' AB. 1 Finding the Circumcenter To ﬁnd a location in a New Vocabulary concurrent point of concurrency 2 Real-World Connection backyard for the largest • • circumcenter of a triangle circumscribed about 3 Finding Lengths of Medians possible swimming pool, • • as in Example 2 • incenter of a triangle • inscribed in 4 Identifying Medians and • median of a triangle • centroid • altitude of a triangle Altitudes • orthocenter of a triangle

Math Background

The theorems in this lesson can be 1 Properties of Bisectors related to Ceva’s Theorem, which Giovanni Ceva published in 1678: Let sides AB , AC , and BC of ABC be divided at X, Y, and Z respec- Hands-On Activity: Paper Folding Bisectors tively. Then AZ , BY , and CX are • Draw and cut out ﬁve different triangles: two acute, two right, and concurrent if and only if one obtuse. AX BZ CY = XB ??ZC YA 1. Concurrency theorems will be • Step 1: Use paper folding to create the angle applied later to inscribed and bisectors of each angle of an acute triangle. circumscribed circles and to the What do you notice about the angle bisectors? study of centroids in physics. • Step 2: Repeat Step 1 with a right triangle and an obtuse triangle. Does your discovery Folding an Angle More Math Background: p. 256C from Step 1 still hold true? Bisector

1. The bisectors of the ' 1. Make a conjecture about the bisectors of the angles of a triangle. Lesson Planning and of a k meet at a point See Resources k inside the . left. • Step 3: Use paper folding to create the See p. 256E for a list of the perpendicular bisector of each side of an acute resources that support this lesson. triangle. What do you notice about the perpendicular bisectors?

PowerPoint Folding a Perpendicular • Step 4: Repeat Step 3 with a right triangle. 2. The # bis. of the sides of Bisector What do you notice? Bell Ringer Practice a k intersect at a point that might fall inside, 2. Make a conjecture about the perpendicular bisectors of the sides of Check Skills You’ll Need outside, or on the k. a triangle. See left. For intervention, direct students to: Constructing Perpendicular Bisectors Lesson 1-7: Example 3 Extra Skills, Word Problems, Proof 272 Chapter 5 Relationships Within Triangles Practice, Ch. 1 Constructing Angle Bisectors Lesson 1-7: Example 5 Special Needs L1 Below Level L2 Extra Skills, Word Problems, Proof For Example 2, have students copy the diagram. Using Have students use a compass or algebra to confirm Practice, Ch. 1 a compass, have them choose other centers and draw that point (2, 3) is the center of the circle that circles as large as possible that lie within the triangle. contains points O, P, and S in Example 1. They will not find a larger circle. 272 learning style: tactile learning style: tactile When three or more lines intersect in one point, they areconcurrent. The point at which they intersect is thepoint of concurrency. For any triangle, four different 2. Teach sets of lines are concurrent. Theorems 5-6 and 5-7 tell you about two of them.

Guided Instruction Key Concepts Theorem 5-6 The perpendicular bisectors of the sides of a triangle are concurrent at a point Hands-On Activity equidistant from the vertices. Students may construct the angle bisectors and perpendicular Theorem 5-7 bisectors using techniques they The bisectors of the angles of a triangle are concurrent at a point equidistant learned in Lesson 1-7. from the sides. Teaching Tip When discussing Theorems 5-6 and 5-7, emphasize that the point You will prove these theorems in the exercises. of concurrency is equidistant from vertices for perpendicular # bisectors and equidistant from This ﬁgure shows QRS with the S QC ϭ SC ϭ RC sides for angle bisectors. perpendicular bisectors of its sides concurrent at C. The point of concurrency Connection of the perpendicular bisectors of a triangle 1 EXAMPLE to Algebra Vocabulary Tip is called the circumcenter of the triangle. C Remind students that the equation The preﬁx circum is Latin Points Q, R, and S are equidistant from C, Q for “around” or “about.” R of a horizontal line is y = a and the circumcenter. The circle is the equation of a vertical line circumscribed about the triangle. is x = a.

1 EXAMPLE Finding the Circumcenter 2 EXAMPLE Diversity

Coordinate Geometry Find the center y Remember that some students have little or no experience of the circle that you can circumscribe P about #OPS. with houses that have yards big enough to hold a swimming pool. Two perpendicular bisectors of (2, 3) y ϭ 3 # = sides of OPS are x 2 and PowerPoint y = 3. These lines intersect at (2, 3). This point is the center Additional Examples OSx of the circle. ϭ x 2 1 Find the center of the circle that circumscribes XYZ. (3, 4) Quick Check 1 a. Find the center of the circle that you can circumscribe about the triangle with vertices (0, 0), (-8, 0), and (0, 6). (–4, 3) y Y b. Critical Thinking In Example 1, explain why it is not necessary to ﬁnd the third perpendicular bisector. Thm. 5-6: All of the # bis. of the sides of a k are concurrent.

This ﬁgure shows #UTV with the ϭ ϭ T XI YI ZI Z bisectors of its angles concurrent at I. X Y O x The point of concurrency of the angle X bisectors of a triangle is called the I incenter of the triangle. Points X, Y, and Z are equidistant from I, U Z the incenter. The circle is inscribed in V the triangle.

Lesson 5-3 Concurrent Lines, Medians, and Altitudes 273

Advanced Learners L4 English Language Learners ELL Have students investigate Ceva’s Theorem and how it The terms circumscribe and inscribe can be related to can be used to prove Theorem 5-8. their prefixes: circum- meaning around and in- meaning within. Students also need to understand the difference between collinear and concurrent.

learning style: verbal learning style: verbal 273 PowerPoint EXAMPLE Real-World Connection Additional Examples 2 Pools The Jacksons want to install the largest possible circular pool in their 2 City planners want to locate a triangular backyard. Where would the largest possible pool be located? fountain equidistant from three straight roads that enclose a park. Locate the center of the pool at the point of concurrency of the angle bisectors. Explain how they can find the This point is equidistant from the sides of the yard. If you choose any other point location. as the center of the pool, it will be closer to at least one of the sides of the yard, and the pool will be smaller. Mariposa Boulevard Quick Check 2 a. The towns of Adamsville, Brooksville, and 2a. Draw segments Cartersville want to build a library that is Adamsville ? connecting the towns. equidistant from the three towns. Trace the Build the library at the Brooksville Park diagram and show where they should build Highway 101 inters. pt. of the # the library. See left. Cartersville Andover Road bisectors of the b. What theorem did you use to ﬁnd the location? segments. The # bisectors of the sides of a k are Locate the fountain at the concurrent at a point equidistant from the vertices. point of concurrency of the angle bisectors of the triangle 12 Medians and Altitudes formed by the three roads.

Amedian of a triangle is a segment Median whose endpoints are a vertex and the Guided Instruction midpoint of the opposite side.

Tactile Learners Students can use paper-folding Key Concepts Theorem 5-8 techniques to find altitudes and D The medians of a triangle are concurrent at a medians of triangles here and in H point that is two thirds the distance from each Exercise 25. C vertex to the midpoint of the opposite side. G E Connection to Physical Science 2 2 2 DC = 3 DJ EC = 3 EG FC = 3 FH J Have students read the Dorling D E B C F 1 A D E B C 2 A D E Kindersly (DK) Activity Lab on B C 3 A D E B C 4 A D E B C 5 A D E pages 302–303, and do the B C Test-Taking Tip Activity involving the centroid as a point of balance. If you don’t remember In a triangle, the point of concurrency of the medians is thecentroid. The point the meaning of a term, is also called the center of gravity of a triangle because it is the point where a like centroid, the Teaching Tip diagram may give a triangular shape will balance. (See DK Activity Lab, page 303.) You will prove The proofs of Theorems 5-8 and clue. Theorem 5-8 in Chapter 6. 5-9 are postponed until students have the tools necessary to 3 EXAMPLE Finding Lengths of Medians complete them. 3 Gridded Response # In ABC at the left, D is the centroid 18 3 EXAMPLE Math Tip and DE = 6. Find BE. / / B . . . . 2 1 = = 0 0 0 Point out that another way to Since D is a centroid, BD 3BE and DE 3 BE. 1 1 1 1 state Theorem 5-8 is that each D 1 2 2 2 2 BE = DE median is broken into segments 3 3 3 3 3 1 4 4 4 4 that have a ratio of 2 : 1. This can A C BE = 6 Substitute 6 for DE. 5 5 5 5 E 3 help students use mental math 6 6 6 6 BE = 18 7 7 7 7 to find lengths. Ask: If BD = 22, 8 8 8 8 what does DE equal? 11 9 9 9 9 Quick Check 3 Find BD. Check that BD + DE = BE. 12

274 Chapter 5 Relationships Within Triangles

274 EXAMPLE Analtitude of a triangle is the perpendicular segment from a vertex to the line 4 EXAMPLE Error Prevention containing the opposite side. Unlike angle bisectors and medians, an altitude of a Students may think that ST and triangle can be a side of a triangle or it may lie outside the triangle. UW meet at the centroid or orthocenter of VSU. Point out that since ST is an altitude and UW is a median, their point of intersection cannot be categorized. For: Concurrent Lines Activity Use: Interactive Textbook, 5-3 Acute Triangle: Right Triangle: Obtuse Triangle: PowerPoint Altitude is inside. Altitude is a side. Altitude is outside. Additional Examples

4 EXAMPLE Identifying Medians and Altitudes 3 M is the centroid of WOR, and WM = 16. Find WX. ST S Is a median, an altitude, or neither? Explain. W ST is a segment extending from vertex S W ST ' VU. to the side opposite S. Also, Z Y ST # V U is an altitude of VSU. T M Quick Check 4 UW O R 4 Is a median, an altitude, or neither? Explain. X Median; UW is a segment drawn from vertex U to the midpt. of 24 the opp. side. 4 The lines containing the altitudes of a triangle are concurrent at the Is KX a median, an altitude, orthocenter of the triangle. A proof of this theorem appears in Chapter 6. neither, or both? K

Key Concepts Theorem 5-9 The lines that contain the altitudes of a triangle are concurrent.

LM X both EXERCISES For more exercises, see Extra Skill, Word Problem, and Proof Practice. Practice and Problem Solving Resources • Daily Notetaking Guide 5-3 L3 • Daily Notetaking Guide 5-3— A Practice by Example Coordinate Geometry Find the center of the circle that you can circumscribe Adapted Instruction L1 Example 1 about each triangle. (page 273) 1.y (–2, –3) 2. y for ᎐ ᎐ 4 GO Help 4 2 O x Closure ᎐2 2 A ᎐4 ᎐4 ᎐22O x ᎐6 (0, 0)

BC Coordinate Geometry Find the center of the circle that you can circumscribe T about kABC. Use the diagram above to explain why the following must be true: 3. A(0, 0) 4. A(0, 0) 5. A(-4, 5) 6. A(-1, -2) 7. A(1, 4) The bisector of the vertex angle B(3, 0) B(4, 0) B(-2, 5) B(-5, -2) B(1, 2) of an isosceles triangle is both C(3, 2) C(4, -3) C(-2, -2) C(-1, -7) C(6, 2) 1 1 an altitude and a median. The 1,1 1 (2, –1 ) (–3, 11 ) (–3, –4 ) (3,1 3) ( 2 ) 2 2 2 2 bisector of the vertex angle Lesson 5-3 Concurrent Lines, Medians, and Altitudes 275 of an isosceles triangle is the perpendicular bisector of the base by Theorem 4-5. Because the bisector is perpendicular, it is an altitude. Because it bisects the opposite side, it is a median.

275 Example 2 Name the point of concurrency of the angle bisectors. 3. Practice (page 274) 8.C 9. Z B Assignment Guide C X 1 AB1-10, 17-19, 21, 24, Y Z 29-31 45° A

2 AB 11-16, 20, 22, 23, 10. City Planning Copy the Altgeld Park 25-28, 32 diagram of Altgeld Park. Show C Challenge 33-36 where park officials should Playground Tennis place a drinking fountain so Court Test Prep 37-41 that it is equidistant from the Mixed Review 42-51 tennis court, the playground, and the volleyball court. # k Homework Quick Check Find the bisectors of the sides of the formed by the Volleyball To check students’ understanding tennis court, the playground, and the volleyball court. Court k of key skills and concepts, go over That point will be equidistant from the vertices of the . Exercises 3, 12, 24, 28, 29. k Example 3 In TUV, Y is the centroid. U TY ≠ 18; TW ≠ 27 (page 274) = Alternative Method 11. If YW 9, find TY and TW. ZY ≠ 4;1 X W 12. If YU = 9, find ZY and ZU. 2 Y Exercise 2 Students may trace ≠ 1 ZU 132 and cut out the triangle and = T V 13. If VX 9, find VY and YX. Z use paper folding, or carefully VY ≠ 6; YX ≠ 3 construct the perpendicular Example 4 Is AB a median, an altitude, or neither? Explain. bisectors on graph paper, to (page 275) B find the point of intersection. 14. 15. 16. 16. Altitude; AB is a A Exercises 3–7 If students use graph segment drawn from paper to draw the triangles, they a vertex of a k perp. A B will easily find the horizontal and to the opp. side. Neither; it’s not a vertical perpendicular bisectors. B segment drawn A Median; A is a midpt. See left. from a vertex. B Apply Your Skills Constructions Draw the triangle. Then construct the inscribed circle and the circumscribed circle. 17–18. See margin. 17. right triangle, #DEF 18. obtuse triangle, #STU

In Exercises 19–22, name each ﬁgure in kBDF. F 19. an angle bisector BE G A 20. a median FC GPS Guided Problem Solving L3 ) E Enrichment L4 21. a perpendicular bisector CA Reteaching L2 22. an altitude DG D B C Adapted Practice L1

PracticeName Class Date L3 23. Critical Thinking A centroid separates a median into two segments. What is the Practice 5-3 Concurrent Lines, Medians, and Altitudes ratio of the lengths of those segments? 1 : 2 or 2 : 1 Find the center of the circle that circumscribes kLMN.

y y y 1. 2. 3. M 4 M 4 M 4 24. Writing Ivars found a yellowed parchment inside an antique book. It read: 2 2 2 L N ؊4 ؊2 24x ؊4 ؊242 6 8 x Ϫ2 2 4 x 24. Find the circumcenter ؊ ؊2 L N From the spot I buried Olaf’s treasure, equal sets of paces did I measure; each 2 ؊4 ؊4 N L of the triangle formed of three directions in a line, there to plant a seedling Norway pine. I could not 4. Construct the angle bisectors for ABC. B Then use the point of concurrency to construct an inscribed circle. by the three pines. return for failing health; now the hounds of Haiti guard my wealth.—Karl A Is AB a perpendicular bisector, an angle bisector, an altitude, a median, C or none of these? After searching Caribbean islands for ﬁve years, Ivars found one with three tall 5.A 6. 7. A A Norway pines. How might Ivars ﬁnd where Karl buried Olaf’s treasure? B

B B

8. 9.A 10. B B 276 Chapter 5 Relationships Within Triangles A

A B

For each triangle, give the coordinates of the point of concurrency of (a) the perpendicular bisectors of the sides and (b) the altitudes.

11.y 12.y 13. y 4 4 4

276 Error Prevention! The ﬁgures below show how to construct medians and altitudes by paper folding. Exercise 10 Students may not realize that the playground and courts locate points. Discuss as a class why these particular points on the playground and courts might have been chosen.

To find an altitude, fold To find a median, fold Then fold so that the fold Exercise 15 Point out that AB Problem Solving Hint the triangle so that a side one vertex to another contains the midpoint and meets only half the conditions overlaps itself and the fold vertex. This locates the the opposite vertex. Paper-folding an to be an altitude and only half altitude is the same as contains the opposite vertex. midpoint of a side. the conditions to be a median, paper-folding the 25–26. Check students’ work. perpendicular to a line 25. Cut out a large triangle. Paper-fold very carefully to construct the three which means that it is neither. through a point not on medians of the triangle and demonstrate Theorem 5-8. the line. Exercise 27 Watch for students 1 26. Cut out a large acute triangle. Paper-fold very carefully to construct the three who think CF is 3GF instead of altitudes of the triangle and demonstrate Theorem 5-9. 2 3GF. Ask: Is CF larger or smaller than GC? larger 27. Multiple Choice C is the centroid of #DEF. If D GF 5 6x2 1 9y , what expression represents CF ? D G Exercises 29, 30 Because using 2 2 2x 1 9y 2x 1 3y C properties from two segments to 6x2 1 9y 4x2 1 6y E prove concurrence is a new and H sophisticated idea, discuss these F proofs as a class after students 28. Is AB a perpendicular bisector, an angle bisector, a median, an altitude, or none complete them. Encourage of these? Explain. b. None of these; it is a midsegment. students to ask questions about Altitude; AB a.A b.A c. A the strategy chosen for each proof. is # to a side from a vertex. Connection to Discrete Math B B Exercise 36 Euler (pronounced l bisector; “oiler”) is also responsible for it bisects an l. B the Seven Bridges of Königsberg problem, the proof of which was 29. Developing Proof Complete this proof of fundamental to the development A GPS Theorem 5-6 by ﬁlling in the blanks. m of graph theory. Have students research Euler’s contributions to O ᐉ n Given: Lines , m, and n are perpendicular mathematics. bisectors of the sides of #ABC. X is 30. It is given that X is X / the intersection of lines / and m. on line and line m. C By the l Bisect. Prove: Line n contains point X, and B Thm., XD 5 XE and XA = XB = XC. 17. AB 5 E XE XF. By the Proof: Since O is the perpendicular bisector of a. 9, XA = XB. Since m is 5 Trans. Prop. of , the perpendicular bisector of b. 9, XB = c. 9. Thus XA = XB = XC. 5 5 XD XE XF. X is Since XA = XC, X is on line n by the Converse of the d. 9 Theorem. on ray n by the Conv. b. BC c. XC d. # bis. of the l Bis. Thm. Proof 30. Prove Theorem 5-7. B ᐉ O F Given: Rays , m, and n are bisectors D F of the angles of #ABC. X is the X E n intersection of rays O and m and GO C nline XD ' AC, XE ' AB, XF ' BC. A D Homework Help m Prove: Ray n contains point X, and XD = XE = XF. 18. Visit: PHSchool.com T Web Code: aue-0503 31. What kind of triangle has its circumcenter on one of its sides? Explain. U A right triangle; check students’ explanations.

Lesson 5-3 Concurrent Lines, Medians, and Altitudes 277

277 32. Coordinate Geometry Complete the following steps to locate the centroid. 4. Assess & Reteach Problem Solving Hint a. Find the coordinates of midpoints L, M, y B (2, 6) You can prove and N. L(1, 3); M(5, 3); N(4, 0) 6 b. See * ) * ) * ) PowerPoint Theorem 5-8 for a b. Find equations of AM ,BN , and CL . general nABC with below 4 c. Find the coordinates of P, the intersection LM Lesson Quiz coordinates A(0, 0), left. * ) * ) of AM and BN . This is the centroid. 10 P B(2b, 2d), and C(2c, 0) * ) ( 3 ,2) 2 by following the steps d. Show that point P is on CL . See left. 1. Complete the sentence: for the particular e. Use the Distance Formula to show that x To find the centroid of a nABC in Exercise 32. 2 26(8, 0) point P is of the distance from each ANC triangle, you need to draw at 3 9 vertex to the midpoint of the opposite side. least median(s). two See margin. 2. FGH has vertices F(–1, 2), G(9, C Challenge For Exercises 33 and 34, points of concurrency have been drawn for two triangles. 2), and H(9, 0). Find 4 Match the points with the lines and segments listed in I–IV. ≠ 3 the center of the circle that 32b. AM: y 5 x; 4 33.A I-D; II-B; III-C; IV-A 34. A circumscribes FGH. (4, 1) BN: y ≠–3x ± 12; 4 Use the diagram for ≠–3 ± 24 C CL: y 7 x 7 B Exercises 3–5. B C –3 10 ±≠–±24 10 V 32d. 7 ()3 7 7 D 24 ≠≠14 2 7 7 D 35. Answers may vary. I-A; II-C; III-B; IV-D Y L X Sample: Let kABC be I. perpendicular bisectors of sides II. angle bisectors isosc. with base ' B N M III. medians IV. lines containing altitudes and C. If AD bisects S P Z lA, then it is # to BC , and therefore the 35. In an isosceles triangle, show that the circumcenter, incenter, centroid, and 3. Identify all medians and altitude* ) from lA. orthocenter can be four different points but all four must be collinear. See left. PX altitudes drawn in PSV. So, AD contains the 36. History In 1765 Leonhard Euler proved that for any triangle, three of the four and SY are medians; VZ is an circumcenter, points of concurrency are collinear. The line that contains these three points is altitude. incenter, centroid, known as Euler’s Line. Use Exercises 33 and 34 to determine which point of 4. If SY = 15, find SM and MY. and orthocenter. concurrency does not necessarily lie on Euler’s Line. l bisectors SM ≠ 10 and MY ≠ 5 5. If MX = 14, find PM and PX. PM ≠ 28 and PX ≠ 42 Test Prep

Alternative Assessment Multiple Choice Use the ﬁgure at the right for Exercises 37–39. 37. What is RD if RL = 54 cm? C W P is a point inside ABC. Have A. 81 cm B. 108 cm students work in pairs to write a C. 162 cm D. 216 cm L D full description of the properties R = of point P if it is the circumcenter, 38. What is WL if WJ 210 mm? H J incenter, centroid, or orthocenter F. 70 mm G. 105 mm of ABC. H. 140 mm J. 157.5 mm F 39. What is x if WL = 15x and LJ = 5x + 3? D A. 0.3 B. 0.4 C. 0.6 D. 1.2 Test Prep Short Response 40. Name all types of triangles for which the centroid, circumcenter, incenter, Resources and orthocenter are all inside the triangle. Classify the triangles according For additional practice with a to the sides as well as the angles. See margin. variety of test item formats: Extended Response 41. The point of concurrency of the three altitudes of a triangle lies outside • Standardized Test Prep, p. 301 the triangle. Where are its circumcenter, incenter, and centroid located in • Test-Taking Strategies, p. 296 relation to the triangle? Draw and label a diagram to support each of • Test-Taking Strategies with your answers. See back of book. Transparencies 278 Chapter 5 Relationships Within Triangles

278 Mixed MixedMixedReview ReviewReview

Use this Checkpoint Quiz to check Lesson 5-2 Determine whether point B must be on the bisector of lT. Explain. GO for students’ understanding of the Help 42. 43. 44. skills and concepts of Lessons 5-1 B See left. B 20 through 5-3. 44. No; point B is not B 25 necessarily equidistant Resources from the sides. T T Grab & Go T S Yes; point B is equidistant • Checkpoint Quiz 1 Yes; TB bisects the l. from the sides. Lesson 3-4 Classify each kJKL by its angles. 45. m&J = 37, m&K = 53, m&L = 90 46. m&J = 47, m&K = 98, m&L = 35 right obtuse Lesson 1-4 In the ﬁgure at the right, ABCD is a square. Identify B C each of the following. 47–51. Answers may vary. * ) * ) * ) * ) A 47. a line skew to ED AB 48. a line skew to EB AD D 49. ABC and ADE 49. two intersecting planes 50. two parallel *segments) 50. AB and CD 51. the intersection of plane ABC and plane BCE BC E 32. e. AM ≠ 34; AP ≠ " 136 ≠ 2 34; Checkpoint Quiz 1 Lessons 5-1 through 5-3 9 3 " Ä BN ≠≠40 2;10 2 " " x Algebra Find the value of x. BP ≠≠160 4 10; 9 3 " 1. 2. 3 Ä 6 12 2x CL ≠ 58; CP ≠ 3 " 232 ≠ 2 58 4x 9 3 " 40. [2]Ä any acute k; or a list 3. a. AB is a midsegment of #XYZ. Y that contains all of AB = 52. Find YZ. 104 the following: b. AX = 26 and BZ = 36. Find A equiangular >, # equilateral >, acute the perimeter of XYZ. 228 Z X B isosceles >, acute scalene > Use the diagram. What can you conclude about each of the following? Explain. [1] a list that does not l 4. &CDB right l; supp. to ADB contain equiangular B > > 5. kABD O kCBD; HL 5. #ABD and #CBD , equilateral , acute isosceles >, or 6. AD and DC > A C scalene AD O DC; CPCTC D S Checkpoint Quiz 7. XY bisects lZXW; ) Use the ﬁgure at the right. 7–8. See left. Z 9. Answers may vary. Y is equidist. from XZ ) 10 Sample: Bisect a side of ) 7. What can you conclude about XY? Explain. X Y a k. Connect the opp. and XW . 21 8. Find XZ. Justify your response. vertex with the midpt. 8. 21; kXYZ OkXYW W by HL, so XZ ≠ 21 by 10. Use the procedure for CPCTC. Writing For a given triangle, describe how you can construct the following. constructing a # to a 9. a median 9–10. See margin. 10. an altitude line from a point not on the line.

lesson quiz, PHSchool.com, Web Code: aua-0503 Lesson 5-3 Concurrent Lines, Medians, and Altitudes 279

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