Geometry

What is BX?

Angle B is 90°. Angle BXA is 90°.

So BX is the height of the Thales circle with radius 5.

$\color{blue}\overline {BX}=5$

 !

A

      5

            X

               5

B                 C

AX =CX  . ..so ....BX  is also an angle  bisector .......so  AB = BC

Triangle AXB ≈  Triangle ABC

AX / AB  = AB / AC

5 / AB = AB / 10

AB^2  = 50

AB = 5sqrt (2) = BC

BX / AB  = BC / AC

BX / AB = AB  / AC

BX / AB = AB / 10

10BX = AB^2

10 BX = 50

BX = 50 / 10    =  5