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Introduction Triangles are typically thought of as simplistic shapes constructed of three angles and three segments. As we continue to explore this shape, we discover there are many more properties and qualities than we may have first imagined. Each property and quality, such as the midsegment of a triangle, acts as a tool for solving problems. 1 1.9.3: Proving the Midsegment of a Triangle

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Page 1: Introduction

IntroductionTriangles are typically thought of as simplistic shapes constructed of three angles and three segments. As we continue to explore this shape, we discover there are many more properties and qualities than we may have first imagined. Each property and quality, such as the midsegment of a triangle, acts as a tool for solving problems.

1

1.9.3: Proving the Midsegment of a Triangle

Page 2: Introduction

Key Concepts• The midpoint is the point on a line segment that

divides the segment into two equal parts.• A midsegment of a triangle is a line segment that joins

the midpoints of two sides of a triangle.

2

1.9.3: Proving the Midsegment of a Triangle

Page 3: Introduction

Key Concepts, continued• In the diagram below, the midpoint of is X.• The midpoint of is Y.• A midsegment of is .

3

1.9.3: Proving the Midsegment of a Triangle

Page 4: Introduction

Key Concepts, continued• The midsegment of a triangle is parallel to the third

side of the triangle and is half as long as the third side. This is known as the Triangle Midsegment Theorem.

4

1.9.3: Proving the Midsegment of a Triangle

Page 5: Introduction

Key Concepts, continued

5

1.9.3: Proving the Midsegment of a Triangle

TheoremTriangle Midsegment Theorem A midsegment of a triangle is parallel to the third side and is half as long.

Page 6: Introduction

Key Concepts, continued• Every triangle has three midsegments.

6

1.9.3: Proving the Midsegment of a Triangle

Page 7: Introduction

Key Concepts, continued• When all three of the midsegments of a triangle are

connected, a midsegment triangle is created.• In the diagram below, .

7

1.9.3: Proving the Midsegment of a Triangle

Page 8: Introduction

Key Concepts, continued• Coordinate proofs, proofs that involve calculations

and make reference to the coordinate plane, are often used to prove many theorems.

8

1.9.3: Proving the Midsegment of a Triangle

Page 9: Introduction

Common Errors/Misconceptions• assuming a segment that is parallel to the third side of

a triangle is a midsegment • incorrectly writing and solving equations to determine

lengths • incorrectly calculating slope • incorrectly applying the Triangle Midsegment Theorem

to solve problems • misidentifying or leaving out theorems, postulates, or

definitions when writing proofs

9

1.9.3: Proving the Midsegment of a Triangle

Page 10: Introduction

Guided PracticeExample 1Find the lengths of BC and YZ and the measure of ∠AXZ.

10

1.9.3: Proving the Midsegment of a Triangle

Page 11: Introduction

Guided Practice: Example 1, continued1. Identify the known information.

Tick marks indicate that X is the midpoint of , Y is the midpoint of , and Z is the midpoint of .

and are midsegments of

11

1.9.3: Proving the Midsegment of a Triangle

Page 12: Introduction

Guided Practice: Example 1, continued2. Calculate the length of BC.

is the midsegment that is parallel to .

The length of is the length of .

12

1.9.3: Proving the Midsegment of a Triangle

Page 13: Introduction

Guided Practice: Example 1, continued

13

1.9.3: Proving the Midsegment of a Triangle

Triangle Midsegment Theorem

Substitute 4.8 for XZ.

Solve for BC.

Page 14: Introduction

Guided Practice: Example 1, continued3. Calculate the measure of YZ. is the midsegment parallel to .

The length of is the length of .

14

1.9.3: Proving the Midsegment of a Triangle

Page 15: Introduction

Guided Practice: Example 1, continued

15

1.9.3: Proving the Midsegment of a Triangle

Triangle Midsegment Theorem

Substitute 11.5 for AB.

Solve for YZ.

Page 16: Introduction

Guided Practice: Example 1, continued4. Calculate the measure of .

16

1.9.3: Proving the Midsegment of a Triangle

Triangle Midsegment Theorem

Alternate Interior Angles Theorem

Page 17: Introduction

Guided Practice: Example 1, continued5. State the answers.

BC is 9.6 units long. YZ is 5.75 units long. m AXZ ∠ is 38°.

17

1.9.3: Proving the Midsegment of a Triangle

✔

Page 18: Introduction

Guided Practice: Example 1, continued

18

1.9.3: Proving the Midsegment of a Triangle

http://www.walch.com/ei/00173
Page 19: Introduction

Guided PracticeExample 3The midpoints of a triangle are X (–2, 5), Y (3, 1), and Z (4, 8). Find the coordinates of the vertices of the triangle.

19

1.9.3: Proving the Midsegment of a Triangle

Page 20: Introduction

Guided Practice: Example 3, continued1. Plot the midpoints on a coordinate plane.

20

1.9.3: Proving the Midsegment of a Triangle

Page 21: Introduction

Guided Practice: Example 3, continued2. Connect the midpoints to form the

midsegments , , and .

21

1.9.3: Proving the Midsegment of a Triangle

Page 22: Introduction

Guided Practice: Example 3, continued3. Calculate the slope of each midsegment.

Calculate the slope of .

The slope of is .22

1.9.3: Proving the Midsegment of a Triangle

Slope formula

Substitute (–2, 5) and (3, 1) for (x1, y1) and (x2, y2).

Simplify.

Page 23: Introduction

Guided Practice: Example 3, continuedCalculate the slope of .

The slope of is 7.

23

1.9.3: Proving the Midsegment of a Triangle

Slope formula

Substitute (3, 1) and (4, 8) for (x1, y1) and (x2, y2).

Simplify.

Page 24: Introduction

Guided Practice: Example 3, continuedCalculate the slope of .

The slope of is .

24

1.9.3: Proving the Midsegment of a Triangle

Slope formula

Substitute (–2, 5) and (4, 8) for (x1, y1) and (x2, y2).

Simplify.

Page 25: Introduction

Guided Practice: Example 3, continued4. Draw the lines that contain the midpoints.

The endpoints of each midsegment are the midpoints of the larger triangle.

Each midsegment is also parallel to the opposite side.

25

1.9.3: Proving the Midsegment of a Triangle

Page 26: Introduction

Guided Practice: Example 3, continued

The slope of is .

From point Y, draw a line that has a slope of .

26

1.9.3: Proving the Midsegment of a Triangle

Page 27: Introduction

Guided Practice: Example 3, continuedThe slope of is 7.

From point X, draw a line that has a slope of 7.

27

1.9.3: Proving the Midsegment of a Triangle

Page 28: Introduction

Guided Practice: Example 3, continued

The slope of is .

From point Z, draw a line that has a slope of

.

The intersections of thelines form the vertices of the triangle.

28

1.9.3: Proving the Midsegment of a Triangle

Page 29: Introduction

Guided Practice: Example 3, continued5. Determine the vertices of the triangle.

The vertices of the triangle are (–3, –2), (9, 4), and (–1, 12), as shown on the following slide.

29

1.9.3: Proving the Midsegment of a Triangle

Page 30: Introduction

Guided Practice: Example 3, continued

30

1.9.3: Proving the Midsegment of a Triangle

✔

Page 31: Introduction

Guided Practice: Example 3, continued

31

1.9.3: Proving the Midsegment of a Triangle

http://www.walch.com/ei/00174

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