We have a triangle $\triangle ABC$ and a point $K$ on $BC$ such that $AK$ is an altitude to $\triangle ABC$. If $AC = 10,$ $BK = 7$, and $BC

If   BC = 13   and   BK = 7  ,  then   KC  =  13 - 7  =  6

And we can use the Pythagorean theorem to find  AK .

CK² + AK²  =  AC²

6² + AK²  =  10²          Subtract  6²  from both sides of this equation.

AK²  =  10² - 6²

AK²  =  64                 Take the positive square root of both sides.

AK  =  8

And let  BC  be the triangle's base, so  AK  is the triangle's height.

area of triangle ABC  =  (1/2) * BC * AK

area of triangle ABC  =  (1/2) * 13 * 8

area of triangle ABC  =  52   sq units