# Midpoint

```Midpoint
Intro
If you measure a line segment that is 10 inches long and divide it into two 5-inch segments, you
get two equal halves. The point at which you divide the line is called the midpoint.
The midpoint of a line segment divides the segment into two congruent parts.
Finding the midpoint of a line segment on a coordinate system is quite simple if the line is
either horizontal or vertical; simply measure the length of the segment and divide by two. For
example,
E
D
A
M
M
B
Graph A
Graph B
In this example, the length of the
vertical segment, AB, is 5 units.
Therefore, the midpoint of AB is 2.5
units from either point.
In this example, the length of the
vertical segment, DE, is 4 units.
Therefore, the midpoint of DE is 2
units from either point.
Namely, the point M(-4 , -3.5).
Namely, the point M(0 , 1).
However, the process becomes a bit more challenging when trying to find the midpoint of a
diagonal segment.
Midpoint Formula
Examining the two graphs more closely, we can determine how the midpoint is being calculated
each time. This will help us in finding a general formula we can use for each scenario.
Graph A shows a line segment, AB, whose coordinates are A(-4,1) and B(-4,-8); the midpoint of
AB is M(-4,-3).
If we think about the general concept of midpoint it is really finding the middle-point or
average of two numbers. Following this thought process, we can see if finding the average of
the x-values and the average of the y-values yields the midpoint:
Midpoint of AB =
=
−4 + −4
1 + −8
and
2
2
−8
−7
and
2
2
= − 4 and − 3 .5
From this calculation, I can see finding the average of the x-values and y-values do indeed yield
the midpoint. Try the same process with the information in graph B.
x +x y +y
The midpoint of a segment with endpoints (x 1 , y 1 ) and (x 2 , y 2 ) is  1 2 , 1 2  .

Applications using the Midpoint Formula
Example 1
Find the coordinates of the center of the rectangle.
Solution
You can find the center of any rectangle by finding
the midpoint of either diagonal.
Using the diagonal AC:
3+1 4 +1
,


2 
 2
4 5
= , 
2 2
= (2 ,2.5 )
2
2

Example 2
Find the midpoint of the hypotenuse for the
right triangle, ΔBCD.
Solution
The hypotenuse of a right triangle is always
the side opposite the right angle; therefore,
the hypotenuse of ΔBCD is BD.
Using BD:
1+3 3+1
,


2 
 2
4 4
= , 
2 2
= (2,2)
Example 3
10 cm
A rectangle is 10 cm long and 7 cm wide. What is
the area of the rhombus formed by connecting
the midpoints of each side of the rectangle?
Solution
7 cm
A rhombus is comprised of two triangles. Draw a
line which connects the midpoints of the lengths
of the rectangle (in green).
Next, we can draw a height for the triangles (in pink). The height of the triangle must be 5cm
since the base connects the midpoints of the length of the rectangle.
1
2
Using our formula for finding the area of a triangle, bh , we can substitute our values:
1
(7)(5) = 17.5 .
2
Finally, since there are two triangles, we can double the area: 17.5(2) = 35 cm2.
```