Numerically, the midpoint of a segment can be considered to be the average of its endpoints. This concept helps in remembering a formula for finding the midpoint of a segment given the coordinates of its endpoints. Recall that the average of two numbers is found by dividing their sum by two.
Theorem 102: If the coordinates of
A and
B are (
x1,
y1) and (
x2,
y2) respectively, then the midpoint,
M, of
AB is given by the following formula
(Midpoint Formula).
Example 1: In Figure
1 ,
R is the midpoint between
Q(−9, −1) and
T(−3, 7). Find its coordinates and use the
Distance Formula to verify that it is in fact the midpoint of
QT
.
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Figure 1
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Finding the coordinates of the midpoint of a line segment.
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By the
Midpoint Formula,
By the
Distance Formula,
Because
QR =
TR and
Q, T, and
R are collinear,
R is the midpoint of
QT
Example 2: If the midpoint of
AB
is (−3, 8) and
A is (12, −1), find the coordinates of
B.
Let the coordinates of
B be (
x, y). Then by the
Midpoint Formula,
Multiply each side of each equation by 2.
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−6 = 12 +
x
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and
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16 = −1 +
y
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−18 =
x
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and
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17 =
y
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Therefore, the coordinates of
B are (−18, 17).
Fundamental Ideas
Coordinate Geometry
