[Properties and characteristics of an isosceles triangle.]

In an isosceles triangle, the center of the inscribed circle divides the height in the ratio $17: 15$. The base is equal to 60. Find the radius of this circle.
Apply the property of the bisector of a triangle.

## Solution

Let $M$ be the base of the altitude $C M$ of the isosceles triangle $A C B (A C=C B)$, and $O$ be the center of the inscribed circle. Then $A M=1 / 2 A B=30$.

Since $A O$ is the bisector of triangle $M A C$, we have $A C: A M=C O: O M$, from which $A C=34$. Therefore, $C M^{2}=A C^{2}-A M^{2}=(34-30)(34+30)=16^{2}$, $O M=15 / 32 C M=15 / 2$

![](https://cdn.mathpix.com/cropped/2024_05_06_14311972f49418f31424g-14.jpg?height=357&width=395&top_left_y=2054&top_left_x=840)

## Answer

7.5 .