Holt McDougal Geometry 5-4 The Triangle Midsegment Theorem Warm Up Use the points A(2, 2), B(12, 2) and C(4, 8) for Exercises 1–5. 1. Find X and Y, the midpoints of AC and CB. 2. Find XY. 3. Find AB. 4. Find the slope of AB. 5. Find the slope of XY. 6. What is the slope of a line parallel to 3x + 2y = 12?

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

Holt McDougal Geometry 5-4 The Triangle Midsegment Theorem 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.

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

Holt McDougal Geometry 5-4 The Triangle Midsegment Theorem Example 2A: Using the Triangle Midsegment Theorem Find each measure. BD = 8.5 ∆ Midsegment Thm. Substitute 17 for AE. Simplify. BD

Holt McDougal Geometry 5-4 The Triangle Midsegment Theorem Example 2B: Using the Triangle Midsegment Theorem Find each measure. mCBD ∆ Midsegment Thm. Alt. Int.  s Thm. Substitute 26° for m  BDF. mCBD = mBDF mCBD = 26°

Holt McDougal Geometry 5-4 The Triangle Midsegment Theorem The positions of the longest and shortest sides of a triangle are related to the positions of the largest and smallest angles.

Holt McDougal Geometry 5-4 The Triangle Midsegment Theorem Example 2A: Ordering Triangle Side Lengths and Angle Measures Write the angles in order from smallest to largest. The angles from smallest to largest are F, H and G. The shortest side is, so the smallest angle is F. The longest side is, so the largest angle is G.

Holt McDougal Geometry 5-4 The Triangle Midsegment Theorem Example 2B: Ordering Triangle Side Lengths and Angle Measures Write the sides in order from shortest to longest. mR = 180° – (60° + 72°) = 48° The smallest angle is R, so the shortest side is. The largest angle is Q, so the longest side is. The sides from shortest to longest are

Holt McDougal Geometry 5-4 The Triangle Midsegment Theorem A triangle is formed by three segments, but not every set of three segments can form a triangle.

Holt McDougal Geometry 5-4 The Triangle Midsegment Theorem A certain relationship must exist among the lengths of three segments in order for them to form a triangle.

Holt McDougal Geometry 5-4 The Triangle Midsegment Theorem Example 3A: Applying the Triangle Inequality Theorem Tell whether a triangle can have sides with the given lengths. Explain. 7, 10, 19 No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths.

Holt McDougal Geometry 5-4 The Triangle Midsegment Theorem Example 4: Finding Side Lengths The lengths of two sides of a triangle are 8 inches and 13 inches. Find the range of possible lengths for the third side.

Holt McDougal Geometry 5-4 The Triangle Midsegment Theorem Assignment Pg. 336 (11-26)