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Prove and use properties of triangle midsegments.

Objective. Prove and use properties of triangle midsegments. 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 and use properties of triangle midsegments.

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  1. Objective Prove and use properties of triangle midsegments.

  2. A midsegment of a triangleis a segment that joins the midpoints of two sides of the triangle. Every triangle has three midsegments, which form the midsegment triangle.

  3. 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 . Example 1: Examining Midsegments in the Coordinate Plane Step 1 Find the coordinates of M and N.

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

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

  6. The vertices of ΔRST are R(–7, 0), S(–3, 6), and T(9, 2). M is the midpoint of RT, and N is the midpoint of ST. Show that and Check It Out! Example 1 Step 1 Find the coordinates of M and N.

  7. Step 2 Compare the slopes of MN and RS. Since the slopes are equal . Check It Out! Example 1 Continued

  8. Step 3 Compare the heights of MN and RS. The length of MN is half the length of RS. Check It Out! Example 1 Continued

  9. The relationship shown in Example 1 is true for the three midsegments of every triangle.

  10. Example 2A: Using the Triangle Midsegment Theorem Find each measure. BD ∆ Midsegment Thm. Substitute 17 for AE. BD = 8.5 Simplify.

  11. Example 2B: Using the Triangle Midsegment Theorem Find each measure. mCBD ∆ Midsegment Thm. mCBD = mBDF Alt. Int. s Thm. mCBD = 26° Substitute 26° for mBDF.

  12. Check It Out! Example 2a Find each measure. JL ∆ Midsegment Thm. Substitute 36 for PN and multiply both sides by 2. 2(36) = JL 72 = JL Simplify.

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

  14. 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

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