# The midpoint M of AB has coordinates (4,9). If the coordinates of A are (2,8), what are the coordinates of B?

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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;

4(+2), 9(+1)

**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)**

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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

so

(2+x)/2=4

multiply both sides by 2 and you get

(2+x)=8

x=6, you do the exact same thing with the y coordinates

(8+y)/2=9

(8+y)=18

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.