$\left[\begin{array}{l}\text { [Mean proportionals in a right triangle] } \\ {[\quad \underline{\text { Pythagorean Theorem (direct and inverse). }}]}\end{array}\right]$

In a right triangle $ABC$, a height $BD$ is dropped from the vertex $B$ of the right angle to the hypotenuse $AC$. It is known that $AB=13, BD=12$.

Find the area of triangle $ABC$.
According to the Pythagorean theorem from triangle $A B D$, we find that $A D=5$.

Since $A C={ }^{A B^{2} /}{ }_{A D}=169 / 5$, then $S_{A B C}=1 / 2 A C \cdot B D=202.8$.

![](https://cdn.mathpix.com/cropped/2024_05_06_0ef0b80dacb6d98eff5ag-34.jpg?height=363&width=851&top_left_y=710&top_left_x=612)

## Answer

202.8.