[ Pythagorean Theorem (direct and inverse). ] [ Auxiliary similar triangles ]

## [ Properties and criteria of tangents $\quad]$

Circle inscribed in an angle $\quad]$

A circle is constructed on the base of an isosceles triangle, which is equal to 8, as a chord, and is tangent to the lateral sides of the triangle.

Find the radius of the circle if the height dropped to the base of the triangle is 3.
Consider similar triangles.

## Solution

Let $B C$ be the base of the given isosceles triangle $A B C$, and $O$ be the center of the circle. Then $A O \perp B C$ and $B M=M C$, where $M$ is the intersection point of segments $A O$ and $B C$.

From the right triangle $A M C$, we find that $A C=5$.

From the similarity of triangles $M C O$ and $M A C$, it follows that $O C: A C=M C: A M$. Therefore, $O C=A C / A M \cdot M C=$ $20 / 3$.

![](https://cdn.mathpix.com/cropped/2024_05_06_d3cd76ad0dc53ae543bdg-28.jpg?height=360&width=435&top_left_y=1864&top_left_x=820)

## Answer

$20 / 3$