Given a sector equal to a quarter of a circle with radius $R$. Find the length of the tangent line drawn at the midpoint of its arc to the intersection with the extensions of the extreme radii of the sector.
Let $O$ be the center of the circle, $M$ be the point of tangency. Prove that the ray $O M$ forms angles of $45^{\circ}$ with the radii bounding the given sector.

## Solution

Let $O$ be the center of the circle, rays $O A$ and $O B$ form a right angle and intersect the given tangent at points $A$ and $B$, and $M$ be the point of tangency. Then

$$
\angle M O B=\angle M O A=\frac{1}{2} \angle A O B=45^{\circ}
$$

Therefore,

$$
M B=M O=R, M A=M O=R
$$

Thus, $A B=2 R$.

## Answer

$2 R$