holt mcdougal geometry 5-4 the triangle midsegment theorem warm up use the points a(2, 2), b(12, 2)...

Holt McDougal Geometry

5-4 The Triangle Midsegment Theorem

Warm UpUse 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 TheoremExample 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 TheoremExample 2B: Using the Triangle Midsegment Theorem

Find each measure.

mCBD

∆ Midsegment Thm.Alt. Int. s Thm.

Substitute 26° for mBDF.

mCBD = mBDF mCBD = 26°

Holt McDougal Geometry

5-4 The Triangle Midsegment TheoremThe 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 TheoremExample 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 TheoremExample 2B: Ordering Triangle Side Lengths and Angle

Measures Write the sides in order from shortest to longest.mR = 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 TheoremExample 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 TheoremExample 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)