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THE MIDPOINT FORMULA
Jump right to the exercises!
The midpoint of the line segment between points
(
x
1
,
y
1
)
and
(
x
2
,
y
2
)
is
(
x
1
+
x
2
2
,
y
1
+
y
2
2
)
.
Here,
x
1
(read as "x sub 1") denotes the x-coordinate of the first point,
and
y
1
(read as "y sub 1")
denotes the y-coordinate of the first point.
Similarly,
x
2
denotes the x-value of the second point,
and
y
2
denotes the y-value of the second point.
Thus, to find the location that is exactly halfway between two points,
you average the x-values, and average the y-values.
The Midpoint Formula follows easily from the following observations:
- The average of two numbers always lies exactly halfway between the two numbers.
- ΔABD
is similar to
ΔAMC
(see below).
That is, these two triangles have the same angles.
-
Since
AM¯
is exactly half of
AB¯
,
AC¯
must be exactly half of
AD¯
.
-
Similarly,
DE¯
must be exactly half of
DB¯
.
EXAMPLES:
Question: Find the midpoint of the line segment between (1,-3) and (-2,5).
Solution:
(
1
+
(
-
2
)
2
,
(
-
3
)
+
5
2
)
=
(
-
1
2
,
1
)
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.