Midpoints (Geometry 2_5)

MidpointsMidpoints

You will learn to find the coordinates of the midpoint of a segment.

BA C

MidpointsMidpoints

You will learn to find the coordinates of the midpoint of a segment.

The midpoint of a line segment, , is the point C that ______ the segment.AB

BA C

MidpointsMidpoints

You will learn to find the coordinates of the midpoint of a segment.

bisectsThe midpoint of a line segment, , is the point C that ______ the segment.AB

BA C

MidpointsMidpoints

You will learn to find the coordinates of the midpoint of a segment.

bisectsThe midpoint of a line segment, , is the point C that ______ the segment.AB

BA C

-7 -6 -5 -4 -2-3 0-1 1 2 3 4 5 6 7

A BC

MidpointsMidpoints

You will learn to find the coordinates of the midpoint of a segment.

bisectsThe midpoint of a line segment, , is the point C that ______ the segment.AB

BA C

-7 -6 -5 -4 -2-3 0-1 1 2 3 4 5 6 7

A BC

C = [3 + (-5)] ÷ 2

MidpointsMidpoints

You will learn to find the coordinates of the midpoint of a segment.

bisectsThe midpoint of a line segment, , is the point C that ______ the segment.AB

BA C

-7 -6 -5 -4 -2-3 0-1 1 2 3 4 5 6 7

A BC

C = [3 + (-5)] ÷ 2

= (-2) ÷ 2

MidpointsMidpoints

You will learn to find the coordinates of the midpoint of a segment.

bisectsThe midpoint of a line segment, , is the point C that ______ the segment.AB

BA C

-7 -6 -5 -4 -2-3 0-1 1 2 3 4 5 6 7

A BC

C = [3 + (-5)] ÷ 2

= (-2) ÷ 2

= -1

MidpointsMidpoints

Theorem 2 – 5

On a number line, the coordinate of the midpoint of a segmentwhose endpoints have coordinates a and b is

MidpointsMidpoints

Theorem 2 – 5

On a number line, the coordinate of the midpoint of a segmentwhose endpoints have coordinates a and b is

2

a b

MidpointsMidpoints

Theorem 2 – 5

On a number line, the coordinate of the midpoint of a segmentwhose endpoints have coordinates a and b is

2

a b

A B

2

a b

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Find the midpoint, C(x, y), of a segment on the coordinate plane.

A

B

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Find the midpoint, C(x, y), of a segment on the coordinate plane.

Consider the x-coordinate: x = 1 x = 9

It will be (midway) between the lines

x = 1 and x = 9 A

B

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Find the midpoint, C(x, y), of a segment on the coordinate plane.

Consider the x-coordinate: x = 1 x = 9

It will be (midway) between the lines

x = 1 and x = 9

x

A

B

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Find the midpoint, C(x, y), of a segment on the coordinate plane.

Consider the x-coordinate: x = 1 x = 9

y = 7

y = 3

It will be (midway) between the lines

x = 1 and x = 9

Consider the y-coordinate:

It will be (midway) between the lines

y = 3 and y = 7

x

A

B

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Find the midpoint, C(x, y), of a segment on the coordinate plane.

Consider the x-coordinate: x = 1 x = 9

y = 7

y = 3

It will be (midway) between the lines

x = 1 and x = 9

Consider the y-coordinate:

It will be (midway) between the lines

y = 3 and y = 7

x

y

A

B

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Find the midpoint, C(x, y), of a segment on the coordinate plane.

Consider the x-coordinate: x = 1 x = 9

y = 7

y = 3

It will be (midway) between the lines

x = 1 and x = 9

Consider the y-coordinate:

It will be (midway) between the lines

y = 3 and y = 7

A

B

x

yC(x, y)

MidpointsMidpoints

Theorem 2 – 6

On a coordinate plane, the coordinates of the midpoint of a

segment whose endpoints have coordinates (x1, y1) and (x2, y2) are

MidpointsMidpoints

Theorem 2 – 6

On a coordinate plane, the coordinates of the midpoint of a

segment whose endpoints have coordinates (x1, y1) and (x2, y2) are

1 2 1 2,2 2

x x y y

MidpointsMidpoints

Theorem 2 – 6

On a coordinate plane, the coordinates of the midpoint of a

segment whose endpoints have coordinates (x1, y1) and (x2, y2) are

1 2 1 2,2 2

x x y y

O

y

x

1 1( , )x y

2 2( , )x y

MidpointsMidpoints

Theorem 2 – 6

On a coordinate plane, the coordinates of the midpoint of a

segment whose endpoints have coordinates (x1, y1) and (x2, y2) are

1 2 1 2,2 2

x x y y

O

y

x

1 1( , )x y

2 2( , )x y

1 2 1 2,2 2

x x y y

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Find the midpoint, C(x, y), of a segment on the coordinate plane.

,C

A(1, 7)

B(9, 3)

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Find the midpoint, C(x, y), of a segment on the coordinate plane.

x = 1 x = 9

x

,C

1 2

2

x x

A(1, 7)

B(9, 3)

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Find the midpoint, C(x, y), of a segment on the coordinate plane.

x = 1 x = 9

y = 7

y = 3

x

y

,C

1 2

2

y y1 2

2

x x

A(1, 7)

B(9, 3)

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Find the midpoint, C(x, y), of a segment on the coordinate plane.

x = 1 x = 9

y = 7

y = 3

A(1, 7)

B(9, 3)

x

y

,C

1 2

2

y y1 2

2

x x

,C

7 3

2

1 9

2

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Find the midpoint, C(x, y), of a segment on the coordinate plane.

x = 1 x = 9

y = 7

y = 3

A(1, 7)

B(9, 3)

x

y

,C

1 2

2

y y1 2

2

x x

,C

7 3

2

1 9

2

,C

10

2

10

2

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Find the midpoint, C(x, y), of a segment on the coordinate plane.

x = 1 x = 9

y = 7

y = 3

A(1, 7)

B(9, 3)

x

yC(5, 5)

,C

1 2

2

y y1 2

2

x x

,C

7 3

2

1 9

2

,C

10

2

10

2

5,5C

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Graph A(1, 1) and B(7, 9)

Draw AB

B(7, 9)

A(1, 1)

Estimate the midpointof AB.

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Graph A(1, 1) and B(7, 9)

Draw AB

B(7, 9)

A(1, 1)

Estimate the midpointof AB.

Check your answerusing the midpoint formula.

1 2 1 2,2 2

x x y yC

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Graph A(1, 1) and B(7, 9)

Draw AB

B(7, 9)

A(1, 1)

Estimate the midpointof AB.

Check your answerusing the midpoint formula.

1+7 1+9,

2 2C

1 2 1 2,2 2

x x y yC

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Graph A(1, 1) and B(7, 9)

C

Draw AB

B(7, 9)

A(1, 1)

Estimate the midpointof AB.

Check your answerusing the midpoint formula.

C(4, 5)

1+7 1+9,

2 2C

8 10,

2 2C

1 2 1 2,2 2

x x y yC

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Suppose C(3, 5) is the midpoint of AB. Find the coordinate of B.

A(7, 2)

C(3, 5)

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Suppose C(3, 5) is the midpoint of AB. Find the coordinate of B.

A(7, 2)

B(x, y) is somewhere over there.

midpoint

C(3, 5)

MidpointsMidpoints

x-coordinate of B

1 2 32

x x

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

y-coordinate of B

1 2 52

y y

Suppose C(3, 5) is the midpoint of AB. Find the coordinate of B.

A(7, 2)

B(x, y) is somewhere over there.

midpoint

C(3, 5)

MidpointsMidpoints

x-coordinate of B

1 2 32

x x

273

2

x

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

y-coordinate of B

1 2 52

y y

225

2

y

Replace x1 with 7and y1 with 2

Suppose C(3, 5) is the midpoint of AB. Find the coordinate of B.

A(7, 2)

B(x, y) is somewhere over there.

midpoint

C(3, 5)

MidpointsMidpoints

x-coordinate of B

1 2 32

x x

273

2

x

27 6x

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

y-coordinate of B

1 2 52

y y

225

2

y

22 10y

Replace x1 with 7and y1 with 2

Multiply each side by 2

Suppose C(3, 5) is the midpoint of AB. Find the coordinate of B.

A(7, 2)

B(x, y) is somewhere over there.

midpoint

C(3, 5)

MidpointsMidpoints

x-coordinate of B

1 2 32

x x

273

2

x

27 6x

2 1x

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

y-coordinate of B

1 2 52

y y

225

2

y

22 10y

2 8y

Replace x1 with 7and y1 with 2

Multiply each side by 2

Add or subtract toisolate the variable

Suppose C(3, 5) is the midpoint of AB. Find the coordinate of B.

A(7, 2)

B(x, y) is somewhere over there.

midpoint

C(3, 5)

MidpointsMidpoints

x-coordinate of B

1 2 32

x x

273

2

x

27 6x

2 1x

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

y-coordinate of B

1 2 52

y y

225

2

y

22 10y

2 8y

Replace x1 with 7and y1 with 2

Multiply each side by 2

Add or subtract toisolate the variable

Suppose C(3, 5) is the midpoint of AB. Find the coordinate of B.

A(7, 2)

B(x, y) is somewhere over there.

B(-1, 8)

midpoint

C(3, 5)

MidpointsMidpoints