Draw a line segment in a coordinate plane. What do you think the coordinates for the midpoint are? Use the Pythagorean Theorem to find the length.
See solution.
Practice makes perfect
We will examine the midpoint and the length separately.
Midpoint
To find the midpoint of a segment, we can start by examining the distance between the endpoints in the x- and y-direction separately. As an example, we could use the line segment between the points A(2,2) and B(8,4).
We can now find the midpoint of AB by identifying the midpoint of the vertical and horizontal segments.
This gives us the coordinates for the midpoint of AB. In this case, it is (5,3).
Length
We will use the same segment as above to demonstrate how to find the length. To find the length of AB, we again use the distances in the x- and y-directions. They are 6 and 2 units, respectively.
The three line segments create a right triangle where AB is the hypotenuse. Therefore, we can use the Pythagorean Theorem to find the length. The lengths of the legs are 6 and 2 units, as stated earlier, so we can substitute a=6 and b=2 to solve for c.
Since c is a length it can never be negative, which is why we discarded the negative square root, -sqrt(40). The length of AB is, therefore, approximately 6.3 units.
