b The line segments between the endpoints and the midpoints are congruent.
A
a See solution.
B
b See solution.
Practice makes perfect
a If the endpoints of a line segment are (x_1,y_1) and (x_2,y_2), we can find the coordinates of the midpoint using the Midpoint Formula.
(x_1+x_2/2,y_1+y_2/2)
This means that to find the x-coordinate of the midpoint, we add the x-coordinates of the endpoints and then divide by 2. Similarly, to find the y-coordinate, we add the y-coordinates of the endpoints and divide by 2.
An Example
If our friend still does not understand why this works, they should think about it in the way shown in the graph below. Let's say that we want to find the midpoint of AB. First, we need to look at the change in the horizontal and vertical directions between the two points.
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).
b We can approach this problem both graphically or algebraically.
Graphically
Let's say A is an endpoint and M is a midpoint between A and B.
In this case, the distance in the x-direction is 3 units and, in the y-direction, is 1 unit.
Since M is the midpoint, MB and AM are congruent. To find the other endpoint we draw an identical line segment, starting at M.
We will then find the other endpoint. In conclusion, we can draw a segment from the midpoint that has the same x-distance and y-distance as the given endpoint to the midpoint.
Algebraically
We can also examine the Midpoint Formula. If one of the endpoints is given by (x_1,y_1) and the midpoint is given by (x_M,y_M), the relation between the x-coordinates is as follows.
x_M=x_1+x_2/2
Here, x_2 is the x-coordinate for the other endpoint. Now, we can solve for x_2.
For the y-coordinate of the second endpoint, the method is the same. We'd get y_2=2y_M-y_1.
We can always find the coordinates of the second endpoint using these equations.
( x_2, y_2) = ( 2x_M-x_1, 2y_M-y_1)
