Assignment

• P. 298-301: 1-6, 8, 10, 12-19 some, 20, 21, 24, 29, 30, 36, 37, 47, 48

• Challenge Problems

Warm-Up

A midsegment of a triangle is a segment that connects the midpoints of two sides of the triangle. Every triangle has 3 midsegments.

Warm-Up

1. On a piece of patty paper, draw a large acute ΔABC.

2. Find the midpoints of each side by putting two vertices on top of each other and pinching the midpoint.

Warm-Up

3. Label the midpoints M, N, and P. Draw the three midsegments of your triangle by connecting the midpoints of each side.

Warm-Up

4. Use another piece of patty paper to trace off ΔAMP.

P

M

A

Warm-Up

5. Compare all the small triangles. What do you notice about the length of a midsegment and the opposite side of the triangle? What kind of lines do they appear to be?

P

M

A

Warm-Up

5. Compare all the small triangles. What do you notice about the length of a midsegment and the opposite side of the triangle? What kind of lines do they appear to be?

P

M

A

Warm-Up

5. Compare all the small triangles. What do you notice about the length of a midsegment and the opposite side of the triangle? What kind of lines do they appear to be?

P

M

A

5.1 Midsegment Theorem and Coordinate Proof

Objectives:

1. To discover and use the Midsegment Theorem

2. To write a coordinate proof

Midsegment

A midsegment of a triangle is a segment that connects the midpoints of two sides of the triangle. Every triangle has 3 midsegments.

Midsegment

A midsegment of a triangle is a segment that connects the midpoints of two sides of the triangle. Every triangle has 3 midsegments.

Example 1

Graph ΔACE with coordinates A(-1, -1), C(3, 5), and E(7, -5). Graph the midsegment MS that connects the midpoints of AC and CE.

6

4

2

-2

-4

5

6

4

2

-2

-4

5

E

C

A

6

4

2

-2

-4

5

E

C

A

6

4

2

-2

-4

5

E

C

A

6

4

2

-2

-4

5

S

M

E

C

A

6

4

2

-2

-4

5

S

M

E

C

A

Example 1

Now find the slope and length of MS and AE. What do you notice about the midsegment and the third side of the triangle?

6

4

2

-2

-4

5

S

M

E

C

A

Midsegment Theorem

The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side.

Example 2

The diagram shows an illustration of a roof truss, where UV and VW are midsegments of ΔRST. Find UV and RS.

Example 3

1.

2.

Deep, Penetrating Questions

How many examples did we look at to come up with our Theorem?

Is that enough?

How could we prove this theorem?

Where could we prove this theorem?

Coordinate Proof

Coordinate proofs are easy. You just have to conveniently place your geometric figure in the coordinate plane and use variables to represent each vertex.– These variables, of course, can represent any

and all cases.– When the shape is in the coordinate plane,

it’s just a simple matter of using formulas for distance, slope, midpoints, etc.

Example 4

Place a rectangle in the coordinate plane in such a way that it is convenient for finding side lengths. Assign variables for the coordinates of each vertex.

Example 4

Convenient placement usually involves using the origin as a vertex and lining up one or more sides of the shape on the x- or y-axis.

Example 5

Place a triangle in the coordinate plane in such a way that it is convenient for finding side lengths. Assign variables for the coordinates of each vertex.

Example 6

Place the figure in the coordinate plane in a convenient way. Assign coordinates to each vertex.

1. Right triangle: leg lengths are 5 units and 3 units

2. Isosceles Right triangle: leg length is 10 units

Example 7

A square has vertices (0, 0), (m, 0), and (0, m). Find the fourth vertex.

y

x

0, m

m, 0 0, 0

y

x

m, m 0, m

m, 0 0, 0

Example 8

Find the missing coordinates. The show that the statement is true.

Example 9

Write a coordinate proof for the Midsegment Theorem.

y

x

O

W

L

y

x

SM

O

W

L

y

x

b, c

a, 0 0, 0

SM

O

W

L

Given: MS is a midsegment of ΔOWL

Prove: MS || OL and MS = ½OL

Example 10

Explain why the choice of variables below might be slightly more convenient.

Given: MS is a midsegment of ΔOWL

Prove: MS || OL and MS = ½OL

y

x

2b, 2c

2a, 0 0, 0

SM

O

W

L

Assignment

• P. 298-301: 1-6, 8, 10, 12-19 some, 20, 21, 24, 29, 30, 36, 37, 47, 48

• Challenge Problems