The altitude x bisects the isosceles triangle into two congruent right triangles,
one leg 10/2=5, one leg x, hypotenuse 13.
We have
5² + x² = 13²
I'll skip the solving and the square root; we recognize the (5,12,13) Pythagorean Triple, the second one on the list after (3,4,5).
Answer: x=12
