[ Two tangents drawn from one point ] [ Rectangles and squares. Properties and criteria ]

A circle of radius $R$ is inscribed in a right angle, touching the sides of the angle at points $A$ and $B$. A tangent is drawn through some point on the smaller arc $A B$ of the circle, cutting off a triangle from the given angle. Find the perimeter of this triangle.
Apply the theorem of equality of tangent segments drawn from one point to a circle.

## Solution

Let a circle with center $O$ be inscribed in the right angle $A C B$ ( $A$ and $B$ - points of tangency), and a line tangent to the circle at point $M$, lying on the smaller arc $A B$, intersects the sides $C A$ and $C B$ of the angle $A C B$ at points $P$ and $Q$ respectively. Then the quadrilateral $A O B C$ is a square. Therefore,

$$
\begin{gathered}
C P+P Q+Q C=C P+(P M+M Q)+Q C=(C P+P M)+(M Q+Q C)= \\
=(C P+P A)+(M Q+Q B)=C A+C B=R+R=2 r
\end{gathered}
$$

A tangent is drawn to a circle inscribed in a square with side $a$, intersecting two of its sides. Find the perimeter of the cut-off triangle.

## Hint

Apply the theorem of equality of tangent segments drawn from one point to a circle.

## Answer

a.