5 4 The Triangle Midsegment Theorem Objective Prove

5 -4 The Triangle Midsegment Theorem Objective Prove and use properties of triangle midsegments. Holt Mc. Dougal Geometry

5 -4 The Triangle Midsegment Theorem _________ – a perpendicular segment from a vertex to the line containing the opposite side. Every triangle has three altitudes. An altitude can be inside, outside, or on the triangle. Holt Mc. Dougal Geometry

5 -4 The Triangle Midsegment Theorem ________– a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. Every triangle has three medians. Holt Mc. Dougal Geometry

5 -4 The Triangle Midsegment Theorem __________– a segment that joins the midpoints of two sides of the triangle. Every triangle has three midsegments, which form the midsegment triangle. Holt Mc. Dougal Geometry

5 -4 The Triangle Midsegment Theorem 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. Holt Mc. Dougal Geometry

5 -4 The Triangle Midsegment Theorem Example 1 Continued Step 2 Compare the slopes of MN and XY. Holt Mc. Dougal Geometry

5 -4 The Triangle Midsegment Theorem Example 1 Continued Step 3 Compare the heights of MN and XY. Holt Mc. Dougal Geometry

5 -4 The Triangle Midsegment Theorem The relationship shown in Example 1 is true for the three midsegments of every triangle. Holt Mc. Dougal Geometry

5 -4 The Triangle Midsegment Theorem Example 2 a Find each measure. JL Holt Mc. Dougal Geometry

5 -4 The Triangle Midsegment Theorem Example 2 b Find each measure. PM Holt Mc. Dougal Geometry

5 -4 The Triangle Midsegment Theorem Example 2 c Find each measure. m MLK Holt Mc. Dougal Geometry

5 -4 The 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? Holt Mc. Dougal Geometry