Question

In the figure given below O is the centre of the circle and \[CD=DE=EF=FG\] if \[\angle COD={{40}^{\circ }}\] then the value of reflex \[\angle COG\]
(a) \[{{200}^{\circ }}\]
(b) \[{{220}^{\circ }}\]
(c) \[{{250}^{\circ }}\]
(d) \[{{280}^{\circ }}\]

Answer 1

Hint: We solve this problem by using the standard condition of chords of the circle.
We have the condition that the angles made at the center by equal chords are equal to each other.
For the following figure

If \[PQ=RS\] then \[\angle POQ=\angle SOR\]
Then we use the condition that reflex of \[\theta \] is given as \[{{360}^{\circ }}-\theta \]
We are given that in the given figure
\[CD=DE=EF=FG\]
We know the condition that the angles made at the center by equal chords are equal to each other.
For the following figure

Complete step by step answer:
If \[PQ=RS\] then \[\angle POQ=\angle SOR\]
By using the above condition to given figure we have
\[\Rightarrow \angle COD=\angle DOE=\angle EOF=\angle FOG\]
Here we can see that the angle \[\angle COG\] is divided in to 4 angles such that the sum of all angles is equal to \[\angle COG\]
By using the above condition to \[\angle COG\] we get
\[\begin{align}
  & \Rightarrow \angle COG=\angle COD+\angle DOE+\angle EOF+\angle FOG \\
 & \Rightarrow \angle COG=\angle COD+\angle COD+\angle COD+\angle COD \\
 & \Rightarrow \angle COG=4\angle COD \\
\end{align}\]
We are given that the angle
\[\angle COD={{40}^{\circ }}\]
By substituting the given angle in the above equation we get
\[\begin{align}
  & \Rightarrow \angle COG=4\times \left( {{40}^{\circ }} \right) \\
 & \Rightarrow \angle COG={{160}^{\circ }} \\
\end{align}\]
We are asked to find the value of reflex \[\angle COG\]
We know that the condition that reflex of \[\theta \] is given as \[{{360}^{\circ }}-\theta \]

By using the above formula to \[\angle COG\] we get
\[\begin{align}
  & \Rightarrow ref\left( \angle COG \right)={{360}^{\circ }}-\angle COG \\
 & \Rightarrow ref\left( \angle COG \right)={{360}^{\circ }}-{{160}^{\circ }} \\
 & \Rightarrow ref\left( \angle COG \right)={{200}^{\circ }} \\
\end{align}\]
Therefore we can conclude that the value of reflex \[\angle COG\] is \[{{200}^{\circ }}\]
So, option (a) is the correct answer.

Note:
Students may make mistakes without finding the reflex angle.
We have the value of \[\angle COG\] as
\[\Rightarrow \angle COG={{160}^{\circ }}\]
But we are asked to find the reflex angle then we get
\[\begin{align}
  & \Rightarrow ref\left( \angle COG \right)={{360}^{\circ }}-\angle COG \\
 & \Rightarrow ref\left( \angle COG \right)={{360}^{\circ }}-{{160}^{\circ }} \\
 & \Rightarrow ref\left( \angle COG \right)={{200}^{\circ }} \\
\end{align}\]
But students may leave the solution at \[\angle COG={{160}^{\circ }}\]
This is not the correct answer for the question.
The reflex angle is the angle that is more than \[{{180}^{\circ }}\] but the angle \[\angle COG={{160}^{\circ }}\] is less than \[{{180}^{\circ }}\] so we use the condition that that reflex of \[\theta \] is given as \[{{360}^{\circ }}-\theta \] to find the required answer.