Section 1-3Distance and Midpoints

Wednesday, September 28, 2011

Essential Questions

How do you find the distance between two points?

How do you find the midpoint of a segment?

Wednesday, September 28, 2011

Vocabulary

1. Pythagorean Theorem:

2. Distance:

Wednesday, September 28, 2011

Vocabulary

1. Pythagorean Theorem: a2 + b2 = c2 , where a and

b are legs of a right triangle, and c is the hypotenuse

2. Distance:

Wednesday, September 28, 2011

Vocabulary

1. Pythagorean Theorem: a2 + b2 = c2 , where a and

b are legs of a right triangle, and c is the hypotenuse

2. Distance: The length of the segment formed between two points

Wednesday, September 28, 2011

Vocabulary

1. Pythagorean Theorem: a2 + b2 = c2 , where a and

b are legs of a right triangle, and c is the hypotenuse

2. Distance: The length of the segment formed between two points

d = (x

2− x

1)2 + (y

2− y

1)2 for (x

1, y

1) and (x

2, y

2)

Wednesday, September 28, 2011

Vocabulary

3. Midpoint:

4. Segment Bisector:

Wednesday, September 28, 2011

Vocabulary

3. Midpoint: The point on a segment that is halfway between the endpoints

4. Segment Bisector:

Wednesday, September 28, 2011

Vocabulary

3. Midpoint: The point on a segment that is halfway between the endpoints

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟ for (x

1, y

1) and (x

2, y

2)

4. Segment Bisector:

Wednesday, September 28, 2011

Vocabulary

3. Midpoint: The point on a segment that is halfway between the endpoints

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟ for (x

1, y

1) and (x

2, y

2)

4. Segment Bisector: Any segment, line, or plane that intersects another segment at its midpoint

Wednesday, September 28, 2011

Example 1Use the number line to find DJ.

−4 −3−2 −1 0 1 2 3 4

J D

Wednesday, September 28, 2011

Example 1Use the number line to find DJ.

−4 −3−2 −1 0 1 2 3 4

J D

x

2− x

1

Wednesday, September 28, 2011

Example 1Use the number line to find DJ.

−4 −3−2 −1 0 1 2 3 4

J D

x

2− x

1 4 − (−4)

Wednesday, September 28, 2011

Example 1Use the number line to find DJ.

−4 −3−2 −1 0 1 2 3 4

J D

x

2− x

1 4 − (−4)

4 + 4

Wednesday, September 28, 2011

Example 1Use the number line to find DJ.

−4 −3−2 −1 0 1 2 3 4

J D

x

2− x

1 4 − (−4)

4 + 4

8

Wednesday, September 28, 2011

Example 1Use the number line to find DJ.

−4 −3−2 −1 0 1 2 3 4

J D

x

2− x

1 4 − (−4)

4 + 4

8 8

Wednesday, September 28, 2011

Example 1Use the number line to find DJ.

−4 −3−2 −1 0 1 2 3 4

J D

x

2− x

1 4 − (−4)

4 + 4

8 8

8 units

Wednesday, September 28, 2011

Example 2Graph A(3, 2) and B(6, 8). Then use the

Pythagorean Theorem to find AB.

Wednesday, September 28, 2011

Example 2Graph A(3, 2) and B(6, 8). Then use the

Pythagorean Theorem to find AB.

x

y

Wednesday, September 28, 2011

Example 2Graph A(3, 2) and B(6, 8). Then use the

Pythagorean Theorem to find AB.

x

y

A

Wednesday, September 28, 2011

Example 2Graph A(3, 2) and B(6, 8). Then use the

Pythagorean Theorem to find AB.

x

y

A

B

Wednesday, September 28, 2011

Example 2Graph A(3, 2) and B(6, 8). Then use the

Pythagorean Theorem to find AB.

x

y

A

B

Wednesday, September 28, 2011

Example 2Graph A(3, 2) and B(6, 8). Then use the

Pythagorean Theorem to find AB.

x

y

A

B

3

6

Wednesday, September 28, 2011

Example 2Graph A(3, 2) and B(6, 8). Then use the

Pythagorean Theorem to find AB.

x

y

A

B

3

6 a

2 + b2 = c2

Wednesday, September 28, 2011

Example 2Graph A(3, 2) and B(6, 8). Then use the

Pythagorean Theorem to find AB.

x

y

A

B

3

6 a

2 + b2 = c2

32 + 62 = c2

Wednesday, September 28, 2011

Example 2Graph A(3, 2) and B(6, 8). Then use the

Pythagorean Theorem to find AB.

x

y

A

B

3

6 a

2 + b2 = c2

32 + 62 = c2

9 + 36 = c2

Wednesday, September 28, 2011

Example 2Graph A(3, 2) and B(6, 8). Then use the

Pythagorean Theorem to find AB.

x

y

A

B

3

6 a

2 + b2 = c2

32 + 62 = c2

9 + 36 = c2

45 = c2

Wednesday, September 28, 2011

Example 2Graph A(3, 2) and B(6, 8). Then use the

Pythagorean Theorem to find AB.

x

y

A

B

3

6 a

2 + b2 = c2

32 + 62 = c2

9 + 36 = c2

45 = c2

45 = c2

Wednesday, September 28, 2011

Example 2Graph A(3, 2) and B(6, 8). Then use the

Pythagorean Theorem to find AB.

x

y

A

B

3

6 a

2 + b2 = c2

32 + 62 = c2

9 + 36 = c2

45 = c2

45 = c2

c ≈ 6.708203933 units

Wednesday, September 28, 2011

Example 3Use the distance formula to find the distance

between A(3, 2) and B(6, 8).

Wednesday, September 28, 2011

Example 3Use the distance formula to find the distance

between A(3, 2) and B(6, 8).

d = (x

2− x

1)2 + (y

2− y

1)2

Wednesday, September 28, 2011

Example 3Use the distance formula to find the distance

between A(3, 2) and B(6, 8).

d = (x

2− x

1)2 + (y

2− y

1)2

= (6 − 3)2 + (8 − 2)2

Wednesday, September 28, 2011

Example 3Use the distance formula to find the distance

between A(3, 2) and B(6, 8).

d = (x

2− x

1)2 + (y

2− y

1)2

= (6 − 3)2 + (8 − 2)2

= 32 + 62

Wednesday, September 28, 2011

Example 3Use the distance formula to find the distance

between A(3, 2) and B(6, 8).

d = (x

2− x

1)2 + (y

2− y

1)2

= (6 − 3)2 + (8 − 2)2

= 32 + 62

= 9 + 36

Wednesday, September 28, 2011

Example 3Use the distance formula to find the distance

between A(3, 2) and B(6, 8).

d = (x

2− x

1)2 + (y

2− y

1)2

= (6 − 3)2 + (8 − 2)2

= 32 + 62

= 9 + 36

= 45

Wednesday, September 28, 2011

Example 3Use the distance formula to find the distance

between A(3, 2) and B(6, 8).

d = (x

2− x

1)2 + (y

2− y

1)2

= (6 − 3)2 + (8 − 2)2

= 32 + 62

= 9 + 36

= 45

≈ 6.708203933 unitsWednesday, September 28, 2011

Example 4Find the midpoint of AB for points A(3, 2) and

B(6, 8).

Wednesday, September 28, 2011

Example 4Find the midpoint of AB for points A(3, 2) and

B(6, 8).

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟

Wednesday, September 28, 2011

Example 4Find the midpoint of AB for points A(3, 2) and

B(6, 8).

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟

=

3 + 62

,2 + 8

2

⎛

⎝⎜⎞

⎠⎟

Wednesday, September 28, 2011

Example 4Find the midpoint of AB for points A(3, 2) and

B(6, 8).

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟

=

3 + 62

,2 + 8

2

⎛

⎝⎜⎞

⎠⎟

=

92

,102

⎛

⎝⎜⎞

⎠⎟

Wednesday, September 28, 2011

Example 4Find the midpoint of AB for points A(3, 2) and

B(6, 8).

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟

=

3 + 62

,2 + 8

2

⎛

⎝⎜⎞

⎠⎟

=

92

,102

⎛

⎝⎜⎞

⎠⎟ =

92

,5⎛

⎝⎜⎞

⎠⎟

Wednesday, September 28, 2011

Example 4Find the midpoint of AB for points A(3, 2) and

B(6, 8).

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟

=

3 + 62

,2 + 8

2

⎛

⎝⎜⎞

⎠⎟

=

92

,102

⎛

⎝⎜⎞

⎠⎟ =

92

,5⎛

⎝⎜⎞

⎠⎟ or 4.5,5( )

Wednesday, September 28, 2011

Example 5Find the coordinates of U if F(−2, 3) is the

midpoint of UO and O has coordinates of (8, 6).

Wednesday, September 28, 2011

Example 5Find the coordinates of U if F(−2, 3) is the

midpoint of UO and O has coordinates of (8, 6).

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟

Wednesday, September 28, 2011

Example 5Find the coordinates of U if F(−2, 3) is the

midpoint of UO and O has coordinates of (8, 6).

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟

(−2,3) =

x + 82

,y + 6

2

⎛

⎝⎜⎞

⎠⎟

Wednesday, September 28, 2011

Example 5Find the coordinates of U if F(−2, 3) is the

midpoint of UO and O has coordinates of (8, 6).

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟

(−2,3) =

x + 82

,y + 6

2

⎛

⎝⎜⎞

⎠⎟

−2 =

x + 82

Wednesday, September 28, 2011

Example 5Find the coordinates of U if F(−2, 3) is the

midpoint of UO and O has coordinates of (8, 6).

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟

(−2,3) =

x + 82

,y + 6

2

⎛

⎝⎜⎞

⎠⎟

−2 =

x + 82

i2

Wednesday, September 28, 2011

Example 5Find the coordinates of U if F(−2, 3) is the

midpoint of UO and O has coordinates of (8, 6).

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟

(−2,3) =

x + 82

,y + 6

2

⎛

⎝⎜⎞

⎠⎟

−2 =

x + 82

i2 (2)i( )

Wednesday, September 28, 2011

Example 5Find the coordinates of U if F(−2, 3) is the

midpoint of UO and O has coordinates of (8, 6).

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟

(−2,3) =

x + 82

,y + 6

2

⎛

⎝⎜⎞

⎠⎟

−2 =

x + 82

i2 (2)i( )

−4 = x + 8

Wednesday, September 28, 2011

Example 5Find the coordinates of U if F(−2, 3) is the

midpoint of UO and O has coordinates of (8, 6).

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟

(−2,3) =

x + 82

,y + 6

2

⎛

⎝⎜⎞

⎠⎟

−2 =

x + 82

i2 (2)i( )

−4 = x + 8

x = −12

Wednesday, September 28, 2011

Example 5Find the coordinates of U if F(−2, 3) is the

midpoint of UO and O has coordinates of (8, 6).

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟

(−2,3) =

x + 82

,y + 6

2

⎛

⎝⎜⎞

⎠⎟

−2 =

x + 82

i2 (2)i( )

−4 = x + 8

x = −12

3 =

y + 62

Wednesday, September 28, 2011

Example 5Find the coordinates of U if F(−2, 3) is the

midpoint of UO and O has coordinates of (8, 6).

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟

(−2,3) =

x + 82

,y + 6

2

⎛

⎝⎜⎞

⎠⎟

−2 =

x + 82

i2 (2)i( )

−4 = x + 8

x = −12

3 =

y + 62

i2

Wednesday, September 28, 2011

Example 5Find the coordinates of U if F(−2, 3) is the

midpoint of UO and O has coordinates of (8, 6).

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟

(−2,3) =

x + 82

,y + 6

2

⎛

⎝⎜⎞

⎠⎟

−2 =

x + 82

i2 (2)i( )

−4 = x + 8

x = −12

3 =

y + 62

i2 2i

Wednesday, September 28, 2011

Example 5Find the coordinates of U if F(−2, 3) is the

midpoint of UO and O has coordinates of (8, 6).

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟

(−2,3) =

x + 82

,y + 6

2

⎛

⎝⎜⎞

⎠⎟

−2 =

x + 82

i2 (2)i( )

−4 = x + 8

x = −12

3 =

y + 62

i2 2i

6 = y + 6

Wednesday, September 28, 2011

Example 5Find the coordinates of U if F(−2, 3) is the

midpoint of UO and O has coordinates of (8, 6).

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟

(−2,3) =

x + 82

,y + 6

2

⎛

⎝⎜⎞

⎠⎟

−2 =

x + 82

i2 (2)i( )

−4 = x + 8

x = −12

3 =

y + 62

i2 2i

6 = y + 6

y = 0

Wednesday, September 28, 2011

Example 5Find the coordinates of U if F(−2, 3) is the

midpoint of UO and O has coordinates of (8, 6).

M =x

1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜

⎞

⎠⎟

(−2,3) =

x + 82

,y + 6

2

⎛

⎝⎜⎞

⎠⎟

−2 =

x + 82

i2 (2)i( )

−4 = x + 8

x = −12

3 =

y + 62

i2 2i

6 = y + 6

y = 0 U(−12,0)

Wednesday, September 28, 2011

Example 6Find PQ if Q is the midpoint of PR.

2x + 3 4x − 1P Q R

Wednesday, September 28, 2011

Example 6Find PQ if Q is the midpoint of PR.

2x + 3 4x − 1P Q R

2x + 3 = 4x − 1

Wednesday, September 28, 2011

Example 6Find PQ if Q is the midpoint of PR.

2x + 3 4x − 1P Q R

2x + 3 = 4x − 1

4 = 2x

Wednesday, September 28, 2011

Example 6Find PQ if Q is the midpoint of PR.

2x + 3 4x − 1P Q R

2x + 3 = 4x − 1

4 = 2x

x = 2

Wednesday, September 28, 2011

Example 6Find PQ if Q is the midpoint of PR.

2x + 3 4x − 1P Q R

2x + 3 = 4x − 1

4 = 2x

x = 2

PQ = 2x + 3

Wednesday, September 28, 2011

Example 6Find PQ if Q is the midpoint of PR.

2x + 3 4x − 1P Q R

2x + 3 = 4x − 1

4 = 2x

x = 2

PQ = 2x + 3

PQ = 2(2) + 3

Wednesday, September 28, 2011

Example 6Find PQ if Q is the midpoint of PR.

2x + 3 4x − 1P Q R

2x + 3 = 4x − 1

4 = 2x

x = 2

PQ = 2x + 3

PQ = 2(2) + 3

PQ = 4 + 3

Wednesday, September 28, 2011

Example 6Find PQ if Q is the midpoint of PR.

2x + 3 4x − 1P Q R

2x + 3 = 4x − 1

4 = 2x

x = 2

PQ = 2x + 3

PQ = 2(2) + 3

PQ = 4 + 3

PQ = 7 units

Wednesday, September 28, 2011

Check for Understanding

Take a look at p. 30 #1-12 to see if you know what you would need to do to solve the

problems

Wednesday, September 28, 2011

Problem Set

Wednesday, September 28, 2011

Problem Set

p. 31 #13-55 odd, 68, 69

“Learn from yesterday, live for today, hope for tomorrow. The important thing is not to stop

questioning.” - Albert EinsteinWednesday, September 28, 2011