The midpoint M of AB has coordinates (4,9). If the coordinates of A are (2,8), what are the coordinates of B?
Since M is the midpoint of a linear segment, then by definition the segment from A to M will have the same slope as the segment from M to B.
If we find the slope of AM, and apply it to M, it will lead us to B.
Comparing M and A, we find that there is a +2 change in X, and a +1 change in Y. All we need to do is apply these changes to the coordinates of M;
B = 6,10
The midpoint between two points (a,b) and (c,d) is given by `((a+c)/2, (b+d)/2)`
We have that one point is (2,8). So we can let a=2 and b=8.
We also know the midpoint is (4,9).
From this information we get the following (4,9) =`((2+c)/2,(8+d)/2)`
Since two points are equal only if both coordinates are equal we have:
To solve the second equation, multiply both sides by 2. This gives you 8 = 2+c. Subtract 2 from both sides to find 6 = c.
To solve the first equation, multiply both sides by 2. This gives you 18=8+d. Subtract 8 from both sides to find d = 10.
So the other point is B = (6,10)
the midpoint has co-ordinates (4,9)
and the co-ordinates of A are (2,8)
need to find co-ordinates of B
(x co-ordinate of A +x coordinate of B)/2= x coordinate of midpoint
multiply both sides by 2 and you get
x=6, you do the exact same thing with the y coordinates
y=10 so the coordinates of B are (6,10)
I learned a cool trick in school for problems like this. When you are given one point on a line and then the midpoint of the line, you can set up the problem like this:
Point A: (2,8) Midpoint: (4,9) Point B: (x,y)
If you count the difference between the x-value of Point A (2) and the x-value of the midpoint (4), you get a difference of 2. If you add 2 to the x-value of the midpoint, you get 6. This is the x-value of Point B.
You can do the same for the y-value. The difference between the y-value of Point A (8) and the y-value of the midpoint (9) is 1. Adding 1 to the y-value of the midpoint gives you 10 for the y-value of Point B.
This makes Point B=(6,10). That's a cool little shortcut, which I find much easier than remembering an equation.