Question

In the figure given below is a circle with centre O, length of chord AB is equal to the radius of the circle. Find the measure of each of the following.
(1) $ \angle AOB $
(2) $ \angle ACB $
(3) $ {\rm{arc AB}} $
(4) $ {\rm{arc ACB}} $

Answer 1

Hint: This question is based on geometry. In this question a circle with centre O is given and A, B, C are the three points in the circumference of the circle in such a way that the length of chord AB is equal to the radius of the circle. We have to calculate the value of the angles $ \angle AOB $ , $ \angle ACB $ and the length of the arcs AB and ACB.

Complete step-by-step answer:

Given:
The radius of the circle is $ OA = OB $
And we know that the length of chord AB is equal to the radius of the circle
So,
\[
OA{\rm{ }} = {\rm{ }}OB{\rm{ }}\\
 = {\rm{ }}AB
\]
(1) Since all three sides OA, OB and AB are equal of the triangle $ \Delta OAB $ , therefore the triangle $ \Delta OAB $ is an Equilateral Triangle and each angle in an equilateral triangle is equal and the value of each angle is $ 60^\circ $ .
So,
$
\Rightarrow \angle OAB = \angle OBA\\
 = \angle AOB\\
 = 60^\circ
 $

(2) Now using the geometric property,
The value of the angle $ \angle ACB $ is the half of the angle $ \angle AOB $ .
So,
$
\Rightarrow \angle ACB = \dfrac{{\angle AOB}}{2}\\
\Rightarrow \angle ACB = \dfrac{{60^\circ }}{2}\\
\Rightarrow \angle ACB = 30^\circ
 $

(3) We know that relationship between the angle, arc and radius of the arc is given by –
 $ {\rm{Arc = Angle }} \times {\rm{ Radius}} $
So, for the arc AB -
 $
\Rightarrow {\rm{arc AB = }}\angle {\rm{AOB}} \times {\rm{AB}}\\
\Rightarrow {\rm{arc AB = }}\left( {{\rm{60}}^\circ \times \dfrac{\pi }{{180^\circ }}} \right) \times AB\\
\Rightarrow {\rm{arc AB = }}\dfrac{\pi }{3}AB
 $
(4) Using the relationship between the angle, arc and radius of the arc we have –
For the arc ACB -
 $
\Rightarrow {\rm{arc ACB = }}\angle {\rm{ACB}} \times {\rm{AB}}\\
\Rightarrow {\rm{arc ACB = }}\left( {{\rm{30}}^\circ \times \dfrac{\pi }{{180^\circ }}} \right) \times AB\\
\Rightarrow {\rm{arc ACB = }}\dfrac{\pi }{6}AB
 $
Therefore, the answers are given below –
(1) $ \angle AOB = 60^\circ $
(2) $ \angle ACB = 30^\circ $
(3) $ {\rm{arc AB = }}\dfrac{\pi }{3}AB $
(4) $ {\rm{arc ACB = }}\dfrac{\pi }{6}AB $

Note: It should be noted that the value of the angles is in degrees but we cannot put the angles in degrees in the calculation for the arc length so we have to convert the angles from degrees to radians. The conversion formula used to convert degrees into radians is given by –
 $ 1{\rm{ degrees = }}\dfrac{\pi }{{180^\circ }}{\rm{ radians}} $