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The point halfway between the endpoints of a line segment is called the midpoint.
A midpoint divides a line segment into two equal parts.
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In Coordinate Geometry, there is more than one way to determine the midpoint of a line segment.
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Method 1
You may find the midpoint by simply dividing the length of the segment by 2 and counting that value from either of the endpoints.
Find the midpoints of line segments AB and CD.
The length of line segment AB is 8 (by counting). The midpoint is 4 units from
either endpoint. On the graph, this point is (1,4).
The length of line segment CD is 3 (by counting). The midpoint is 1.5 units
from either endpoint. On the graph, this point is (2,1.5)
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If the line segments are diagonal, more thought must be paid to the solution. When you are finding the coordinates of the midpoint of a segment, you are actually finding the average (mean) of the x-coordinates and the average (mean) of the y-coordinates.
Method 2
0 10 20 30-10-20-30
a b
2
ba
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Midpoint Formula
Midpoint Coordinates =
2,
22121yyxx
The Midpoint Formula works for all line segments: vertical, horizontal or diagonal.
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Example 1 - Find a Midpoint on a Number Line
0 1 2 3 4 5 6-1-2-3-4-5-6
Q R
Method 1
The distance from -3 to 4 is 7. Half of 7 is 3.5, which you add to -3. The midpoint is 0.5.
0 1 2 3 4 5 6-1-2-3-4-5-6
Q R
+7
+3.5
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Example 1 - Find a Midpoint on a Number Line
0 1 2 3 4 5 6-1-2-3-4-5-6
Q R
Method 2
Use the midpoint formula
5.021
243
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Example 2 – Find the Coordinates of a Midpoint
Q = 40
P = -20
TEMPERATURE Find the coordinate of the midpoint PQ
The coordinates of P and Q are -20 and 40. Let M be the midpoint of PQ
10or2
202
4020
M
Find the coordinates of M, the midpoint of PQ, for P(-1, 2) and Q(6, 1)
Let P be (x1, y1) and Q be (x2, y2)
2
3,2
5
2
12,
2
61
2,
22121
M
Myyxx
M
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Example 3 - A Different Problem
Find the Coordinates of an Endpoint
Find the coordinates of X if Y(-2, 2) is the midpoint of XQ, and Q has the coordinates (2, 8)
Method 2
Let Z be (x2, y2) in the Midpoint Formula
28
,22
)2,2( 11 yxY
Write two equations to find the coordinates of x
1
1
1
6
242
22
x
x
x
1
1
1
4
842
82
y
y
y
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A Different Approach to a Different Problem
Find the Coordinates of an Endpoint
Method 1
Find the coordinates of X if Y(-2, 2) is the midpoint of XQ, and Q has the coordinates (2, 8)
First, let’s try to visualize the problem.
This will give you an idea of where X is located.
Y
Q
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Example 3 – Use Algebra to Find Measures
3x + 6
2x + 14
A
B
C
A is the midpoint of segment BC.Find x and the measures of ABand AC.
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1) (7 , 4), (9, −1) 2) (8, −9), (0, 5)
3) (1, −7), (1, −12) 4) (0, 4), (−4, -12)
5) (−4, 2), (2, −3) 6) (5, 9), (−1, 9)
7) (−7, 8), (−2, −9) 8) (2, −11), (−9, 0)
9) (4, −1), (2, −7) 10) (−4, −6), (3, −6)
11) (14, 0), (−7, 5) 12) (14, −8), (12, −1)
13) (−4, 12), (−7, −2)
ExercisesGiven the endpoints of a line segment, find the midpoint
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Given the midpoint and one endpoint of a line segment, find the other endpoint.
1) Endpoint:(−9, −1), midpoint:(8, 14)
2) Endpoint:(10, 12), midpoint:(6, 9)
3) Endpoint:(−8, −10), midpoint:(10, −7)
4) Endpoint:(−11, 9), midpoint:(3, −11)
5) Endpoint:(−2, 7), midpoint:(12, −10)
6) Endpoint:(11, 14), midpoint:(10, 14)
7) Endpoint:(14, −8), midpoint:(5, 8)
8) Endpoint:(−9, 0), midpoint:(10, −7)