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Concurrent Lines, Medians, and Altitudes 273 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.
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