## 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 ...

## A chord of a circle is equal to the radius of the circle.

[question] Question. A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc. [/question] [solution] Solution: In $\triangle O A B$ $A B=O A=O B=$ radius $\therefore \triangle O A B$ is an equilateral triangle. Therefore, each interior angle of this triangle will be of $60^{\circ}$. $\therefore \angle A O B=60^{\circ}$ $\angle \mathrm{ACB}=\frac{1}{2} \angle \mathrm{AOB}=\frac{1}{2}\left(60^{...

## If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D

[question] Question. If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD (see figure 10.25). [/question] [solution] Solution: Let us draw a perpendicular OM on line AD. It can be observed that BC is the chord of the smaller circle and AD is the chord of the bigger circle. We know that perpendicular drawn from the centre of the circle bisects the chord. $\therefore B M=M C \ldots(1)$ And, AM = MD ... (2) On subtracting equa...

## If two equal chords of a circle intersect within the circle

[question] Question. If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords. [/question] [solution] Solution: Let PQ and RS are two equal chords of a given circle and they are intersecting each other at point T. Draw perpendiculars OV and OU on these chords. In $\triangle O V T$ and $\triangle O U T$, OV = OU (Equal chords of a circle are equidistant from the centre) $\angle O V T=\angle ...

## If two equal chords of a circle intersect within the circle

[question] Question. If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord [/question] [solution] Solution: Let PQ and RS be two equal chords of a given circle and they are intersecting each other at point T. Draw perpendiculars OV and OU on these chords. In $\triangle O V T$ and $\Delta O U T$, OV = OU (Equal chords of a circle are equidistant from the centre) $\angle O V T=\angle O U T\left(\righ...

## Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm.

[question] Question. Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord. [solution] Solution: Let the radius of the circle centered at O and O' be 5 cm and 3 cm respectively. $O A=O B=5 \mathrm{~cm}$ $O^{\prime} A=O^{\prime} B=3 \mathrm{~cm}$ OO' will be the perpendicular bisector of chord $A B$. $\therefore A C=C B$ It is given that, $O O^{\prime}=4 \mathrm{~cm}$ Let $O C$ be $x$. Therefore, $O^{\prime} ...

## 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...

## Write True or False: Give reasons for your answers.

[question] Question. Write True or False: Give reasons for your answers. (i) Line segment joining the centre to any point on the circle is a radius of the circle. (ii) A circle has only finite number of equal chords. (iii) If a circle is divided into three equal arcs, each is a major arc. (iv) A chord of a circle, which is twice as long as its radius, is a diameter of the circle. (v) Sector is the region between the chord and its corresponding arc. (vi) A circle is a plane figure. [/question] [s...

## Fill in the blanks

[question] Question. Fill in the blanks (i) The centre of a circle lies in __________ of the circle. (exterior/ interior) (ii) A point, whose distance from the centre of a circle is greater than its radius lies in __________ of the circle. (exterior/ interior) (iii) The longest chord of a circle is a __________ of the circle. (iv) An arc is a __________ when its ends are the ends of a diameter. (v) Segment of a circle is the region between an arc and __________ of the circle. (vi) A circle divid...