Question

A square is inscribed in a circle with centre O. What angle does each side subtend at centre O?
(A) \[{{45}^{\circ }}\]
(B) \[{{60}^{\circ }}\]
(C) \[{{75}^{\circ }}\]
\[(D)\text{ }{{90}^{\circ }}\]

Answer 1

Hint: Here, we are going to use the property of circles which is that if the two chords of a circle are equal then they subtend an equal angle at the centre of the circle. Then we are going to use the property that the sum of angles around a point is equal to \[{{360}^{\circ }}\].

Complete step by step answer:
As we know, all sides of a square are equal. Hence, we can say that \[AB=BC=CD=DA\]. Now, the sides of the square represent chords of the circle. Therefore, all four sides of the inscribed square represent four equal chords of the circle.
As we know, there is a property of circles which states that if the chords of a circle are equal then they subtend an equal angle at the centre of the circle.
Hence, from the diagram, we conclude that \[\angle AOB=\angle BOC=\angle COD=\angle DOA\].
Finally, we know that the sum of angles around a point is equal to \[{{360}^{\circ }}\].
Therefore, \[\angle AOB+\angle BOC+\angle COD+\angle DOA={{360}^{\circ }}\]……...equation (1)
Let us assume that \[\angle AOB=x\]. Hence, all the angles will be equal to x,
\[\angle AOB+\angle BOC+\angle COD+\angle DOA={{360}^{\circ }}\]
Putting all value in equation (1), we get,
\[x+x+x+x={{360}^{\circ }}\]
\[\Rightarrow 4x={{360}^{\circ }}\]
Dividing both sides by 4,
\[x={{90}^{\circ }}\]
Hence, we can say that,
\[\angle AOB=\angle BOC=\angle COD=\angle DOA={{90}^{\circ }}\]
Sides of square subtend \[{{90}^{\circ }}\] at centre O of circle.

So, the correct answer is “Option D”.

Note: An inscribed polygon is a polygon in which all vertices lie on a circle.
The polygon is inscribed in the circle and the circle is circumscribed about the polygon.
A circumscribed polygon is a polygon in which each side is a tangent to a circle.
The circle is inscribed in the polygon and the polygon is circumscribed about the circle. (It is a circle in a polygon). Students should be careful in understanding the difference between inscribed and circumscribe. Students should learn all the properties of circles and draw diagrams carefully.