Objective 5-1 Prove and use properties of triangle midsegments. The Triangle Midsegment Theorem Objective Prove and use properties of triangle midsegments. Holt Geometry

5-1 A midsegment of a triangle is a segment that joins the midpoints of two sides of the triangle. Every triangle has three midsegments, which form the midsegment triangle.

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Prove the Midsegment Theorem 5-1 Prove the Midsegment Theorem Write a coordinate proof of the Midsegment Theorem for one midsegment. GIVEN : DE is a midsegment of OBC. PROVE : DE OC and DE = OC 1 2 SOLUTION STEP 1 Place OBC and assign coordinates. Because you are finding midpoints, use 2p, 2q, and 2r. Then find the coordinates of D and E. D( ) ) 2q+0, 2r+0 2 = D(q,r E( ) 2q+2p,2r+0 E(q+p, r)

Prove the Midsegment Theorem STEP 2 5-1 Prove the Midsegment Theorem STEP 2 Prove DE OC . The y-coordinates of D and E are the same, so DE has a slope of 0. OC is on the x-axis, so its slope is 0. Because their slopes are the same, DE OC . STEP 3 Prove DE = OC. Use the Ruler Postulate 1 2 to find DE and OC . DE = (q+p) –q = p OC = 2p–0 = 2p So, the length of DE is half the length of OC

Example 1: Examining Midsegments in the Coordinate Plane 5-1 Example 1: Examining Midsegments in the Coordinate Plane The vertices of ∆XYZ are X(–1, 8), Y(9, 2), and Z(3, –4). M and N are the midpoints of XZ and YZ. Show that and . X (-1, 8) Y (9, 2) Z (3, -4)

Example 1: Examining Midsegments in the Coordinate Plane 5-1 Example 1: Examining Midsegments in the Coordinate Plane The vertices of ∆XYZ are X(–1, 8), Y(9, 2), and Z(3, –4). M and N are the midpoints of XZ and YZ. Show that and . Step 1 Find the coordinates of M and N. X (-1, 8) Y (9, 2) Z (3, -4)

Step 2 Compare the slopes of MN and XY. 5-1 Example 1 Continued Step 2 Compare the slopes of MN and XY. Since the slopes are the same,

Step 3 Compare the heights of MN and XY. 5-1 Example 1 Continued Step 3 Compare the heights of MN and XY.

Place a figure in a coordinate plane 5-1 Place a figure in a coordinate plane Place each figure in a coordinate plane in a way that is convenient for finding side lengths. Assign coordinates to each vertex. a. A rectangle b. A scalene triangle SOLUTION It is easy to find lengths of horizontal and vertical segments and distances from (0, 0), so place one vertex at the origin and one or more sides on an axis.

Place a figure in a coordinate plane a. 5-1 Place a figure in a coordinate plane a. Let h represent the length and k represent the width. b. Notice that you need to use three different variables.

Example 2: Using the Triangle Midsegment Theorem 5-1 Example 2: Using the Triangle Midsegment Theorem Find BD. ∆ Midsegment Thm. Substitute 17 for AE. BD = 8.5 Simplify. Find mCBD. ∆ Midsegment Thm. mCBD = mBDF Alt. Int. s Thm. mCBD = 26° Substitute 26° for mBDF.

Check It Out! Find JL. 2(36) = JL 72 = JL Find mMLK. Find PM. 5-1 Check It Out! Find JL. ∆ Midsegment Thm. Substitute 36 for PN and multiply both sides by 2. 2(36) = JL 72 = JL Simplify. Find mMLK. Find PM. ∆ Midsegment Thm. ∆ Midsegment Thm. mMLK = mJMP Substitute 97 for LK. Similar triangles mMLK = 102° Substitute. PM = 48.5 Simplify.

Use the diagram for Items 1–3. Find each measure. 1. ED 2. AB 3. mBFE 5-1 Lesson Quiz: Part I Use the diagram for Items 1–3. Find each measure. 1. ED 2. AB 3. mBFE 10 14 44°

Lesson Quiz: Part II 4. Find the value of n. 5-1 Lesson Quiz: Part II 4. Find the value of n. 5. ∆XYZ is the midsegment triangle of ∆WUV. What is the perimeter of ∆XYZ? 16 11.5