Midpoint Formula
Andrew Gloag
Anne Gloag
Eve Rawley
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Printed: November 5, 2012
AUTHORS
Andrew Gloag
Anne Gloag
Eve Rawley
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C ONCEPT
Concept 1. Midpoint Formula
1
Midpoint Formula
Here you’ll learn how to use the midpoint formula to find the coordinates of the point that is in the middle of the line
segment connecting two given points. You’ll also use that formula to find one endpoint of a line segment given its
other endpoint and its midpoint.
What if you were given the coordinates of two points like (4, 1) and (0, -3)? How could you find the midpoint of the
line segment joining the two points? After completing this Concept, you’ll be able to find the midpoint of any line
segment using the Midpoint Formula.
Watch This
MEDIA
Click image to the left for more content.
CK-12 Foundation: The Midpoint Formula
Watch This
For a graphic demonstration of the midpoint formula, watch this video:
MEDIA
Click image to the left for more content.
.
PatrickJMT: The MidpointFormula
Guidance
In the last concept, you saw how to find the distance between two points. In this concept, you will learn how to find
the point exactly half way between two points.
Example A
Find the coordinates of the point that is in the middle of the line segment connecting the points
Solution
Let’s start by graphing the two points:
1
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We see that to get from point A to point B we move 6 units down and 10 units to the right.
In order to get to the point that is halfway between the two points, it makes sense that we should move half the
vertical distance and half the horizontal distance—that is, 3 units down and 5 units to the right from point A.
The midpoint is M = (−7 + 5, −2 − 3) = (−2, −5).
The Midpoint Formula
We now want to generalize this method in order to find a formula for the midpoint of a line segment.
Let’s take two general points A = (x1 , y1 ) and B = (x2 , y2 ) and mark them on the coordinate plane:
We see that to get from A to B, we move x2 − x1 units to the right and y2 − y1 units up.
1
In order to get to the half-way point, we
need to move x2 −x
units to the right and
2
y2 −y1 x2 −x1
midpoint M is at x1 + 2 , y1 + 2 .
2 y1 +y2
This simplifies to M = x1 +x
. This is the Midpoint Formula:
2 , 2
The midpoint of the line segment connecting the points (x1 , y1 ) and (x2 , y2 ) is
y2 −y1
2
up from point A. Thus the
x1 +x2 y1 +y2
.
2 , 2
It should hopefully make sense that the midpoint of a line is found by taking the average values of the x and y−values
of the endpoints.
2
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Concept 1. Midpoint Formula
Example B
Find the midpoint between the following points.
a) (-10, 2) and (3, 5)
b) (3, 6) and (7, 6)
Solution
x1 +x2 y1 +y2
2 , 2
2+5
5) is −10+3
2 , 2
Let’s apply the Midpoint Formula:
a) the midpoint of (-10, 2) and (3,
b) the midpoint of (3, 6) and (7, 6) is
6+6
3+7
2 , 2
=
−7 7
2 ,2
=
12
10
2, 2
= (−3.5, 3.5)
= (5, 6)
Example C
A line segment whose midpoint is (2, -6) has an endpoint of (9, -2). What is the other endpoint?
Solution
In this problem we know the midpoint and we are looking for the missing endpoint.
The midpoint is (2, -6).
One endpoint is (x1 , x2 ) = (9, −2).
Let’s call the missing point (x, y).
We know that the x−coordinate of the midpoint is 2, so: 2 =
9+x2
2
⇒ 4 = 9 + x2 ⇒ x2 = −5
We know that the y−coordinate of the midpoint is -6, so:
−6 =
−2 + y2
⇒ −12 = −2 + y2 ⇒ y2 = −10
2
The missing endpoint is (-5, -10).
Here’s another way to look at this problem: To get from the endpoint (9, -2) to the midpoint (2, [U+2011]6), we
had to go 7 units left and 4 units down. To get from the midpoint to the other endpoint, then, we would need to go 7
more units left and 4 more units down, which takes us to (-5, -10).
Watch this video for help with the Examples above.
MEDIA
Click image to the left for more content.
CK-12 Foundation: The Midpoint Formula
Vocabulary
• The Midpoint Formula states that the midpoint of the line segment connecting the points (x1 , y1 ) and (x2 , y2 )
is
x1 +x2 y1 +y2
.
2 , 2
3
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Guided Practice
Find the midpoint between the points (4, -5) and (-4, 5).
Solution
Let’s apply the Midpoint Formula:
The midpoint of (4, -5) and (-4, 5)
x1 +x2 y1 +y2
2 , 2
−5+5
is 4−4
2 , 2
=
0 0
2, 2
= (0, 0)
Practice
Find the midpoint of the line segment joining the two points.
1.
2.
3.
4.
5.
6.
7.
8.
9.
(3, -4) and (6, 1)
(2, -3) and (2, 4)
(4, -5) and (8, 2)
(1.8, -3.4) and (-0.4, 1.4)
(5, -1) and (-4, 0)
(10, 2) and (2, -4)
(3, -3) and (2, 5)
An endpoint of a line segment is (4, 5) and the midpoint of the line segment is (3, -2). Find the other endpoint.
An endpoint of a line segment is (-10, -2) and the midpoint of the line segment is (0, 4). Find the other
endpoint.
10. Find a point that is the same distance from (4, 5) as it is from (-2, -1), but is not
4