$AOB$ is a sector of a circle with center $ O$ and radius $OA = 10$.
Circle with radius $3$
is inscribed in this sector such that it touches radius $OA$, radius $OB$ and arc $AB$.
Find the length of the chord $AB$.
I don't know where to begin. To calculate the length of $AB$ ,
we'll need the length of perpendicular from $O$ to $AB$.
( then we can use pythagoras theorem to get half of $AB$ and then $AB$) .
But how can I find that?
Best Answer
The distance from the centre of the inscribed circle to point $O$ is $7$, so the angle between $AO$ and the perpendicular line from $O$ to $AB$ is $\theta=\arcsin\left(\frac{3}{7}\right)$, assuming that lines $OA$ and $OB$ are tangents to the inscribed circle.
Thus the length of $AB$ is
$$AB=2\times OA\times\sin(\theta)=2\times10\times\frac{3}{7}=8\frac{4}{7}$$