Objective Prove and use properties of triangle midsegments.

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Presentation transcript:

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.

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.

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

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

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

Check It Out! Example 1 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 Step 1 Find the coordinates of M and N.

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

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

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

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.

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.

Check It Out! Example 2b Find each measure. PM ∆ Midsegment Thm. Substitute 97 for LK. PM = 48.5 Simplify.

Check It Out! Example 2c Find each measure. mMLK ∆ Midsegment Thm. mMLK = mJMP Similar triangles mMLK = 102° Substitute.