Question is :
Consider a circle of unit radius centered at $O$ in the plane. let $AB$ be a chord which makes an angle $\theta$ with the tangent to the circle at $A$ .find the area of triangle $OAB$
What all i could do was is just draw the picture and even in that i am not sure if he mean angle $BAP=\theta$ or $BAQ=\theta$. I am assuming for some time that $BAQ=\theta$
Now, I would have some hope if angle $OAB$ can be calculated from given data in terms of theta then i would use the area formula as
$\text{Area of triangle $OAB= \frac{1}{2}.OA.OB.\sin (\angle AOB)$}$ and as $OA=OB=1$ we would then get
Area of triangle $OAB=\frac{1}{2}.\sin (\angle AOB)$
But i am not sure how to relate $\theta$ with $\angle AOB$.
I am not even sure if there is any way to relate this.
I would be thankful if some one can help me with this.
Thank you.
Best Answer
The angle $(\angle BAQ)$ is a tangent (chord) angle.
Theorem An Angle formed by a chord and a tangent that intersect on a circle is half the measure of the intercepted arc. So $$\theta=\frac{1}{2} m \overset{\frown}{(AB)}$$ from there $(\angle AOB)=2\theta$