The value of x=8 and y=3.
Let's denote the midpoint of segment AB as M. Since AD is the perpendicular bisector of AB, it divides AB into two congruent segments. Let AM and BM represent these segments.
Given that BD=2x−2 and CD=10, we can find the lengths of AM and BM using the Pythagorean theorem. The key observation is that AD is perpendicular to BC, so △ADB and △CDB are right triangles.
Using the Pythagorean theorem in △ADB:
AM^2 +BD^2 =AD^2
And in △CDB:
BM^2 +CD^2 =AD^2
Substitute the given values:
AM^2 +(2x−2)^2 =10^2
BM^2 +10^2 =(3x+4)^2
Solve these equations to find x. Once x is determined, substitute it back into the expression for AB to find y:
AB=9y+31
With the values x=8 and y=3, the given conditions are satisfied.