Geometry Section 1-3

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+x22,y1+y22⎛⎝⎜ ⎞⎠⎟ 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+x22,y1+y22⎛⎝⎜ ⎞⎠⎟ =3+62,2+82⎛⎝⎜ ⎞⎠⎟
=92,102⎛⎝⎜ ⎞⎠⎟ =92,5⎛⎝⎜⎞⎠⎟ 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+x22,y1+y22⎛⎝⎜ ⎞⎠⎟ (−2,3)=x+82,y+62⎛⎝⎜ ⎞⎠⎟
−2=x+82i2(2)i( )−4=x+8x=−12
3=y+62i22i6=y+6y=0U(−12,0)

6. Example 3
Find PQ if Q is the midpoint of PR.
2x + 34x − 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 change13x2−x1132−(−3)132+3
135
53 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 change13y2−y1133−(−3)133+3
136
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 change2 units
Horizontal change53 units
P(−3+53,−3+2)P(−43,−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 change25x2−x1251−(−4)251+4
255
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 change25y2−y1252−625−4
25(4)
85 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 change2 units
Vertical change85 units
F(−4+2,6−85)F(−2,225)