Question

Draw a pair of tangents to a circle of radius $5cm$ which are inclined to each other at an angle of ${60}^o$.

Answer 1

Lets assume a circle with radius =r

and AP and AQ are two external tangents and $20={60}^o$ be the angle between them.

therefore $\angle CAP=\theta$ and $\angle CPA={90}^o$

So in $△CAP,\sin \theta =\sin \left(\frac{{60}^o}{2}\right)=\frac{CP}{CA}$

$\sin \left({30}^o\right)=\frac{5}{CA}$

$CA=10cm$

Now lets construct this pair of tangents.

Construct a circle with centre c and radius $=5cm$

locate a point A, which is $10$ cm for C such that $CA=10cm$

Now locate the midpoint of CA be M so that $CM=MA=5cm$

Now draw a circle with centre M and radius $=CM=5cm$ and assume that

this circle intersects the previous circle at points P and Q and join the points P,A and Q,A

So that PA and QA are our desire tangents such that $\angle PAQ={60}^o$