If you mean the base is 52, and the two angles formed by the base and the two sides are 45 degrees, and the vertex angle is 90:

The line drawn from the base to the vertex is the height of the triangle.

That line is 26 inches high. Because:

The line bisects the base, thus the base of the new triangle formed with the height as one side, and half the base as the base, is 26 ((1/2)*52).

That base is opposite the original vertex angle that is now bisected. It was a 90 degree angle, now it's a 45 degree angle.

The base is opposite one 45 degree angle, and the height is opposite another 45 degree angle. The sides opposite congruent angles are congruent.