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.

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Presentation transcript:

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.

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

Step 3 Compare the heights of MN and XY.

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

Example 2A: Find each measure. BD ∆ Midsegment Thm. Substitute 17 for AE. BD = 8.5 Simplify. mCBD ∆ Midsegment Thm. mCBD = mBDF Alt. Int. s Thm. mCBD = 26° Substitute 26° for mBDF.

Geometry 5.4 Triangle Midsegment Theorem Example 3: Indirect Measurement Application In an A-frame support, the distance PQ is 46 inches. What is the length of the support ST if S and T are at the midpoints of the sides? ∆ Midsegment Thm. Substitute 46 for PQ. ST = 23 Simplify. The length of the support ST is 23 inches.