The Triangle Midsegment Theorem
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Transcript of The Triangle Midsegment Theorem
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Holt McDougal Geometry
The Triangle Midsegment TheoremThe Triangle Midsegment Theorem
Holt Geometry
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt McDougal Geometry
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Holt McDougal Geometry
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?
(3, 5), (8, 5)
5
10
0
0
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Holt McDougal Geometry
The Triangle Midsegment Theorem
Prove and use properties of triangle midsegments.
Objective
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Holt McDougal Geometry
The Triangle Midsegment Theorem
midsegment of a triangle
Vocabulary
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Holt McDougal Geometry
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.
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Holt McDougal Geometry
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 .
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Holt McDougal Geometry
The Triangle Midsegment Theorem
Example 1 Continued
Step 2 Compare the slopes of MN and XY.
Since the slopes are the same,
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Holt McDougal Geometry
The Triangle Midsegment Theorem
Step 3 Compare the heights of MN and XY.
Example 1 Continued
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Holt McDougal Geometry
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
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Holt McDougal Geometry
The Triangle Midsegment Theorem
Check It Out! Example 1 Continued
Since the slopes are equal .
Step 2 Compare the slopes of MN and RS.
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Holt McDougal Geometry
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.
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Holt McDougal Geometry
The Triangle Midsegment Theorem
The relationship shown in Example 1 is true for the three midsegments of every triangle.
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Holt McDougal Geometry
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
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Holt McDougal Geometry
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°
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Holt McDougal Geometry
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
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Holt McDougal Geometry
The Triangle Midsegment Theorem
Check It Out! Example 2b
Find each measure.
PM
∆ Midsegment Thm.
Substitute 97 for LK.
Simplify.PM = 48.5
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Holt McDougal Geometry
The Triangle Midsegment Theorem
Check It Out! Example 2c
Find each measure.
mMLK
∆ Midsegment Thm.
Similar triangles
Substitute.
mMLK = mJMP
mMLK = 102°
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Holt McDougal Geometry
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
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Holt McDougal Geometry
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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Holt McDougal Geometry
The Triangle Midsegment Theorem
Lesson Quiz: Part I
Use the diagram for Items 1–3. Find each measure.
1. ED
2. AB
3. mBFE
10
14
44°
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Holt McDougal Geometry
The Triangle Midsegment Theorem
Lesson Quiz: Part II
4. Find the value of n.
5. ∆XYZ is the midsegment triangle of ∆WUV.
What is the perimeter of ∆XYZ?
16
11.5