Question

The radius of a circle is 30cm. Find the length of the arc of this circle, if the length of the chord of the arc is 30cm.

Answer 1

Hint: Use the fact that in a triangle, equal sides have equal opposite angles. Use the fact that the sum of the angles of a triangle is $180{}^\circ $. Hence find the measure of the angle subtended by the chord at the centre. Hence find the length of the arc.

Complete step-by-step answer:

We have AB = 30cm and AC = 30cm
Hence AB = AC.
Since in a triangle, equal sides have equal opposite angles, we have
$\angle B=\angle C$
Also, we have AB = 30cm and BC = 30cm.
Hence AB = BC.
Since in a triangle, equal sides have equal opposite angles, we have
$\angle A=\angle C$
Hence, we have
$\angle A=\angle B=\angle C$
Now, we have
$\angle A+\angle B+\angle C=180{}^\circ $ angle sum property of a triangle
Since $180{}^\circ =\pi $ radians, we have
$\angle A+\angle A+\angle A=\pi \Rightarrow 3\angle A=\pi $
Hence, we have
$\angle A=\dfrac{\pi }{3}$
Now, we know that if x is the measure of the angle in radians, l is the length of the arc and r is the radius of the circle, then we have $x=\dfrac{l}{r}$
Hence if l is the length of the arc(BC), then we have
$\dfrac{\pi }{3}=\dfrac{l}{30}$
Hence, we have
$l=10\pi $ cm.

Note: Alternative solution: General method.
Let BC = a and the radius be r.

Draw AD perpendicular BC.
Now we have $BD=\dfrac{a}{2}$ and AB = r.
Let $\angle BAD=\theta $
Hence, we have
$\sin \theta =\dfrac{BD}{AB}=\dfrac{a}{2r}$
Hence we have
$\theta =\arcsin \left( \dfrac{a}{2r} \right)$
Hence $\angle BAC=2\theta =2\arcsin \left( \dfrac{a}{2r} \right)$
In the above question, we have a = 30cm and r = 30cm.
Hence, we have
$\angle BAC=2{{\sin }^{-1}}\left( \dfrac{30}{2\times 30} \right)=2{{\sin }^{-1}}\dfrac{1}{2}=2\times \dfrac{\pi }{6}=\dfrac{\pi }{3}$
Hence arc(BC) $=\dfrac{\pi }{3}\times 30=10\pi $ cm