[ Properties and characteristics of an isosceles triangle. ] [ Area of a triangle (using height and base). ]

A circle with its center on side $A C$ of the isosceles triangle $A B C (A B = B C)$ touches sides $A B$ and $B C$. Find the radius of the circle if the area of triangle $A B C$ is 25, and the ratio of the height $B D$ to side $A C$ is $3: 8$.
Let $A D=D C=4 x, B D=3 x$. Then $B C=5 x$. The heights are inversely proportional to the corresponding sides, so the height to the lateral side is $8 / 5 \cdot 3 x=4.8 x$. The radius $r$ of the circle is obviously half of this height.

According to the condition $4 x \cdot 3 x=25$, from which $x=\frac{5}{6} \sqrt{3}$, and $r=\frac{12}{5} x=2 \sqrt{3}$.

## Answer

$2 \sqrt{3}$.