[ Pythagorean Theorem (direct and inverse).]

Two sides of a triangle are 25 and 30, and the height to the third side is 24. Find the third side.
Consider two possible cases.

## Solution

Let in triangle $ABC$, $AB=25$, $AC=30$, and the height $AD=24$. Then $BD^2 = AB^2 - AD^2 = 49$, $CD^2 = AC^2 - AD^2 = 30^2 - 24^2 = 6 \cdot 54 = 18^2$.

Therefore, $BC = BD + DC = 7 + 18 = 25$, if point $D$ is on segment $BC$, and $BC = DC - BD = 18 - 7 = 11$, if outside it.

![](https://cdn.mathpix.com/cropped/2024_05_06_04ea0aec75eb28164af1g-21.jpg?height=300&width=1172&top_left_y=-1&top_left_x=446)

## Answer

25 or 11.