# Find the distance between A(2,5) and B(a,b) . The point m(-1,3) is the midpoint of AB.

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Given the points:

A(2,5) B(a,b) and m(-1,3) is midpoint of AB

We need to determine the point B(a,b)

Then we will use the midpoint formula to solve for B:

We know that:

mx = ( Ax+Bx)/2

==> -1 = ( 2 + a)/2

==> -2 = 2 + a

==>** a = -4**

Also :

my = (yA+yB)/2

==> 3 = (5+b)/2

==? 6 = 5+ b

**==> b= 1**

**Then the point B is: B(-4,1)**

The answer posted above gives the coordinates of the point B, whereas the answer required is the distance between the points A and B.

To find this distance it is not necessary to find the coordinates of B. Since coordinates of A and mid point m of AB are given, we can simply find the distance Am, and double it to get the distance AB.

Solution:

Distance between any two points (x1, y1) and (x2, y20 is given by:

Distance = [(x2 - x1)^2 + (y2 - y)^2]^(1/2)

Therefore:

Distance Am = [(-1 - 2)^2 + (3 - 5)^2]^(1/2)

= (9 + 4)^(1/2) = 13^(1/2)

Distance AB = 2*Distance Am = 2*13^(1/2) = 7.2111