Question

How do you find the length of the chord of a circle with radius $8\,cm$ and a central angle of $110^\circ $.

Answer 1

Hint: First, we will draw a circle and mark the centre as $O$ and the chord as $AB$. Then, we will join the centre and both the ends of the chord and this is our radius which is given as $8\,cm$. Now, we will divide the chord in two equal parts and connect $C$ to the centre of the circle. We will use the trigonometric function $\sin x$ to find the length of the chord.

Complete step by step answer:
In the given question we have to find the chord of a circle of radius $8\,cm$ and having a central angle of $110^\circ $. Chord is defined as the line segment joining any of the two points lying on the circle.
The diagram of the question is below:

According to the diagram we have a chord namely $AB$ which is divided by a line segment $OC$ in two equal halves. So, the central angle is also halved i.e., $55^\circ $.
Now, In triangle $AOC$
We have, $AO = 8\,cm$(radius of a circle)
We know that, $\sin \theta = \dfrac{{perpendicular}}{{hypotenuse}}$
So, in $\vartriangle AOC$
$\sin (55^\circ ) = \dfrac{{AC}}{{AO}}$
Putting the value of $AO$ i.e., $AO = 8\,cm$. We get,
$\sin (55^\circ ) = \dfrac{{AC}}{8}$
Put the value $\sin (55^\circ ) = 0.8191$
$0.8191 = \dfrac{{AC}}{8}$
Therefore, $AC = 8 \times 0.8191$$ = 6.5528$$cm$
As $OC$ divides $AB$ into two equal halves.
So, $AB = 2 \times AC$
$ \Rightarrow AB = (2 \times 6.5528)\,cm$
$ \Rightarrow AB = 13.1056\,cm$
Hence, the length of the chord $AB = 13.1056\,cm$

Note:
We can also solve it by using an alternate method. We will take the triangle $COB$. After this we will put the value of trigonometric function and similarly we find the value of chord $AB$. Note that we have taken $OA$ and $OB$ as radius which is given as $8\,cm$. As radius is defined as the line segment with one end at centre and the other end at anywhere on the circle. So, we termed $OA$ and $OB$ as radius.