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Holt Geometry 5-4 The Triangle Midsegment Theorem midsegment of a triangle Vocabulary

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Page 1: Holt Geometry 5-4 The Triangle Midsegment Theorem midsegment of a triangle Vocabulary

Holt Geometry

5-4 The Triangle Midsegment Theorem

midsegment of a triangle

Vocabulary

Page 2: Holt Geometry 5-4 The Triangle Midsegment Theorem midsegment of a triangle Vocabulary

Holt 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.

Page 3: Holt Geometry 5-4 The Triangle Midsegment Theorem midsegment of a triangle Vocabulary

Holt Geometry

5-4 The Triangle Midsegment Theorem

Example 1: Examining Midsegments in the Coordinate Plane

Step 1 Find the coordinates of M and N.

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 .

Page 4: Holt Geometry 5-4 The Triangle Midsegment Theorem midsegment of a triangle Vocabulary

Holt Geometry

5-4 The Triangle Midsegment Theorem

Example 1 Continued

Step 2 Compare the slopes of MN and XY.

Since the slopes are the same,

Page 5: Holt Geometry 5-4 The Triangle Midsegment Theorem midsegment of a triangle Vocabulary

Holt Geometry

5-4 The Triangle Midsegment Theorem

Step 3 Compare the heights of MN and XY.

Example 1 Continued

Page 6: Holt Geometry 5-4 The Triangle Midsegment Theorem midsegment of a triangle Vocabulary

Holt Geometry

5-4 The Triangle Midsegment Theorem

Check It Out! Example 1

Step 1 Find the coordinates of M and N.

The vertices of ΔRST are R(–7, 0), S(–3, 6), and T(9, 2). M is the midpoint of RT, and N is the midpoint of ST. Show that and

Page 7: Holt Geometry 5-4 The Triangle Midsegment Theorem midsegment of a triangle Vocabulary

Holt Geometry

5-4 The Triangle Midsegment Theorem

Check It Out! Example 1 Continued

Since the slopes are equal .

Step 2 Compare the slopes of MN and RS.

Page 8: Holt Geometry 5-4 The Triangle Midsegment Theorem midsegment of a triangle Vocabulary

Holt Geometry

5-4 The Triangle Midsegment Theorem

Check It Out! Example 1 Continued

The length of MN is half the length of RS.

Step 3 Compare the heights of MN and RS.

Page 9: Holt Geometry 5-4 The Triangle Midsegment Theorem midsegment of a triangle Vocabulary

Holt Geometry

5-4 The Triangle Midsegment Theorem

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

Page 10: Holt Geometry 5-4 The Triangle Midsegment Theorem midsegment of a triangle Vocabulary

Holt 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

Page 11: Holt Geometry 5-4 The Triangle Midsegment Theorem midsegment of a triangle Vocabulary

Holt Geometry

5-4 The Triangle Midsegment Theorem

Example 2B: Using the Triangle Midsegment Theorem

Find each measure.

mCBD

∆ Midsegment Thm.

Alt. Int. s Thm.

Substitute 26° for mBDF.

mCBD = mBDF

mCBD = 26°

Page 12: Holt Geometry 5-4 The Triangle Midsegment Theorem midsegment of a triangle Vocabulary

Holt Geometry

5-4 The Triangle Midsegment Theorem

Check It Out! Example 2a

Find each measure.

JL

∆ Midsegment Thm.

Substitute 36 for PN and multiply both sides by 2.

Simplify.

2(36) = JL

72 = JL

Page 13: Holt Geometry 5-4 The Triangle Midsegment Theorem midsegment of a triangle Vocabulary

Holt Geometry

5-4 The Triangle Midsegment Theorem

Check It Out! Example 2b

Find each measure.

PM

∆ Midsegment Thm.

Substitute 97 for LK.

Simplify.PM = 48.5

Page 14: Holt Geometry 5-4 The Triangle Midsegment Theorem midsegment of a triangle Vocabulary

Holt Geometry

5-4 The Triangle Midsegment Theorem

Check It Out! Example 2c

Find each measure.

mMLK

∆ Midsegment Thm.

Similar triangles

Substitute.

mMLK = mJMP

mMLK = 102°

Page 15: Holt Geometry 5-4 The Triangle Midsegment Theorem midsegment of a triangle Vocabulary

Holt 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?

The length of the support ST is 23 inches.

∆ Midsegment Thm.

Substitute 46 for PQ.

Simplify.ST = 23

Page 16: Holt Geometry 5-4 The Triangle Midsegment Theorem midsegment of a triangle Vocabulary

Holt Geometry

5-4 The Triangle Midsegment Theorem

Check It Out! Example 3

What if…? Suppose Anna’s result in Example 3 (p. 323) is correct. To check it, she measures a second triangle. How many meters will she measure between H and F?

∆ Midsegment Thm.

Substitute 1550 for AE.

HF = 775 m Simplify.


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