Question

Given A(a,2) B(3,4) and C(-2,-1). If A is at the same distance to the points B and C. Find a.

Answer 1

distance AB= sqrt((a-3)^2+(2-4)^2)
distance AC= sqrt(a+2)^2+(2+1)^2)

setting them equal
((a-3)^2+(2-4)^2)=(a+2)^2+(2+1)^2)
expand out the terms, combine, and solve for a.
(a-3)^2-(a+2)^2=9-4
so if the right side is five, then we have on the left a difference of two squares is five, and immediately comes to mind 9 and 4, and because math teachers often chose whole numbers, it is worth a try..
then if (a-3)^2=9, then a must be six, or zero. six wont work, but zero does.

Answer 2

the left side is also just a difference of two squares, so

((a-3)+(a-2))((a-3)-(a-2)) = (2a-5)(-1)
so, you have

5-2a = 5
a = 0