Locating Points and Midpoints

1. Section 1-3
Locating Points and Midpoints

2. Essential Questions
How do you find the midpoint of a segment?
How do you locate a point on a segment given a
fractional distance from one endpoint?

3. Vocabulary
1. Midpoint: The point on a segment that is halfway
between the endpoints
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ for (x1
,y1
) and (x2
,y2
)
2. Segment Bisector: Any segment, line, or plane
that intersects another segment at its midpoint

4. Example 1
Find the midpoint of AB for points A(3, 2) and
B(6, 8).
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ =
3 + 6
2
,
2 + 8
2
⎛
⎝⎜
⎞
⎠⎟
=
9
2
,
10
2
⎛
⎝⎜
⎞
⎠⎟ =
9
2
,5
⎛
⎝⎜
⎞
⎠⎟ or 4.5,5( )

5. Example 2
Find the coordinates of U if F(−2, 3) is the
midpoint of UO and O has coordinates of (8, 6).
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ (−2,3) =
x + 8
2
,
y + 6
2
⎛
⎝⎜
⎞
⎠⎟
−2 =
x + 8
2
i2(2)i ( )
−4 = x + 8
x = −12
3 =
y + 6
2
i22i
6 = y + 6
y = 0
U(−12,0)

6. Example 3
Find PQ if Q is the midpoint of PR.
2x + 3 4x − 1
P Q R
2x + 3 = 4x − 1
4 = 2x
x = 2
PQ = 2x + 3
PQ = 2(2) + 3
PQ = 4 + 3
PQ = 7 units

7. Example 4
Find P if NM that is 1/3 the distance from N to M for
points N(−3, −3) and M(2, 3).
Horizontal change
1
3
x2
− x1
1
3
2 − (−3)
1
3
2 + 3
1
3
5
5
3
units
x
y
M
N

8. Example 4
Find P if NM that is 1/3 the distance from N to M for
points N(−3, −3) and M(2, 3).
Vertical change
1
3
y2
− y1
1
3
3 − (−3)
1
3
3 + 3
1
3
6
2 units
x
y
M
N

9. Example 4
Find P if NM that is 1/3 the distance from N to M for
points N(−3, −3) and M(2, 3).
Vertical change 2 units
Horizontal change
5
3
units
P(−3 +
5
3
,−3 + 2) P(−
4
3
,−1)
x
y
M
N

10. Example 5
Find F on AB such that the ratio of AF to FB is 2:3 for
points A(−4, 6) and B(1, 2).
3:2 means 2 parts of AF and 3 parts of FB for a
total of five parts. To go from A to F, we use 2/5
of the total distance from A to B.

11. Example 5
Find F on AB such that the ratio of AF to FB is 2:3 for
points A(−4, 6) and B(1, 2).
x
y
B
A
Horizontal change
2
5
x2
− x1
2
5
1− (−4)
2
5
1+ 4
2
5
5
2 units

12. Example 5
Find F on AB such that the ratio of AF to FB is 2:3 for
points A(−4, 6) and B(1, 2).
x
y
B
A
Vertical change
2
5
y2
− y1
2
5
2 − 6
2
5
−4
2
5
(4)
8
5
units

13. Example 5
Find F on AB such that the ratio of AF to FB is 2:3 for
points A(−4, 6) and B(1, 2).
x
y
B
A Horizontal change 2 units
Vertical change
8
5
units
F(−4 + 2,6 −
8
5
) F(−2,
22
5
)