## In the given figure, ∠PQR = 100°, where P,

[question] Question. In the given figure, $\angle P Q R=100^{\circ}$, where $P, Q$ and $R$ are points on a circle with centre $O$. Find $\angle O P R$. [/question] [solution] Solution: Consider PR as a chord of the circle. Take any point S on the major arc of the circle. PQRS is a cyclic quadrilateral. $\angle \mathrm{PQR}+\angle \mathrm{PSR}=180^{\circ}$ (Opposite angles of a cyclic quadriateral) $\Rightarrow \angle \mathrm{PSR}=180^{\circ}-100^{\circ}=80^{\circ}$ We know that the angle subtend...

## In the given figure, ∠PQR = 100°,

[question] Question. In the given figure, $\angle P Q R=100^{\circ}$, where $P, Q$ and $R$ are points on a circle with centre $O$. Find $\angle O P R$. [/question] [solution] Solution: Consider PR as a chord of the circle. Take any point S on the major arc of the circle. PQRS is a cyclic quadrilateral. $\angle P Q R+\angle P S R=180^{\circ}$ (Opposite angles of a cyclic quadrilateral) $\Rightarrow \angle \mathrm{PSR}=180^{\circ}-100^{\circ}=80^{\circ}$ We know that the angle subtended by an arc ...

## Suppose you are given a circle.

[question] Question. Suppose you are given a circle. Give a construction to find its centre [solution] Solution: The below given steps will be followed to find the centre of the given circle. Step1. Take the given circle. Step2. Take any two different chords AB and CD of this circle and draw perpendicular bisectors of these chords. Step3. Let these perpendicular bisectors meet at point O. Hence, O is the centre of the given circle....

## Draw different pairs of circles. How many points does each pair have in common?

[question] Question. Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points? [/question] [solution] Solution: Consider the following pair of circles. The above circles do not intersect each other at any point. Therefore, they do not have any point in common. The above circles touch each other only at one point Y. Therefore, there is 1 point in common. The above circles touch each other at 1 point X only. Therefore, the circles ...

## Prove that if chords of congruent circles subtend equal angles at their centres,

[question] Question. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal. [/question] [solution] Solution: Let us consider two congruent circles (circles of same radius) with centres as O and O'. In $\triangle \mathrm{AOB}$ and $\triangle C O^{\prime} D$, $\angle \mathrm{AOB}=\angle \mathrm{CO}^{\prime} \mathrm{D}($ Given $)$ $O A=O^{\prime} C$ (Radii of congruent circles) $O B=O^{\prime} D$ (Radii of congruent circles) \$\therefore \triangle...

## Recall that two circles are congruent if they have the same radii.

[question] Question. Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres [solution] Solution: A circle is a collection of points which are equidistant from a fixed point. This fixed point is called as the centre of the circle and this equal distance is called as radius of the circle. And thus, the shape of a circle depends on its radius. Therefore, it can be observed that if we try to superimpose tw...