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
Objective
Prove and use properties of triangle
midsegments.
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
∆ Midsegment Thm.
Substitute 17 for AE.
BD = 8.5
Simplify.
Holt McDougal Geometry
5-4 The Triangle Midsegment Theorem
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°
Holt McDougal Geometry
Substitute 26° for mBDF.
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 shortest side is
smallest angle is F.
The longest side is
, so the
, so the largest angle is G.
The angles from smallest to largest are F, H and 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)
Holt McDougal Geometry