ABC isosceles triangle AB = 25 cm height CK = 24cm AC base?

In this triangle ABC, consider a part of the triangle, namely triangles AKC, and CKB. Both are rectangular. We apply the theorem to both triangles, denoting AK = x, then KB = (25 – x). We get two equations.

AC ^ 2 = x ^ 2 + KC ^ 2, KB ^ 2 + KC ^ 2 = BC ^ 2,

1) AC ^ 2 = x ^ 2 + 24 ^ 2, 2) (25 – x) ^ 2 + 24 ^ 2 = 25 ^ 2,

(25 – x) ^ 2 = 625 – 576, (25 – x) ^ 2 = 49, 25 – x = √49, x = 25 – 7, x = 18. Now you can substitute the resulting value x = 18 into equation 1 ) to find the desired base AC:

AC ^ 2 = 18 ^ 2 + 24 ^ 2 = 324 + 576 = 900, whence AC = 30.