## In any triangle ABC, if the angle bisector of ∠A and perpendicular bisector of BC intersect,

[question] Question. In any triangle ABC, if the angle bisector of ∠A and perpendicular bisector of BC intersect, prove that they intersect on the circum circle of the triangle ABC. [/question] [solution] Solution: Let perpendicular bisector of side BC and angle bisector of ∠A meet at point D. Let the perpendicular bisector of side BC intersect it at E. Perpendicular bisector of side BC will pass through circumcentre O of the circle. ∠BOC and ∠BAC are the angles subtended by arc BC at the centre...

## Two congruent circles intersect each other at points A and B.

[question] Question. Two congruent circles intersect each other at points A and B. Through A any line segment PAQ is drawn so that P, Q lie on the two circles. Prove that BP = BQ. [/question] [solution] Solution: $\mathrm{AB}$ is the common chord in both the congruent circles. $\therefore \angle \mathrm{APB}=\angle \mathrm{AQB}$ In $\triangle B P Q$, $\angle \mathrm{APB}=\angle \mathrm{AQB}$ $\therefore B Q=B P$ (Angles opposite to equal sides of a triangle) [solution]...

## AC and BD are chords of a circle which bisect each other. Prove that

[question] Question. AC and BD are chords of a circle which bisect each other. Prove that (i) AC and BD are diameters; (ii) ABCD is a rectangle. [/question] [solution] Solution: Let two chords AB and CD are intersecting each other at point O. In $\triangle \mathrm{AOB}$ and $\triangle \mathrm{COD}$,In $\triangle \mathrm{AOB}$ and $\triangle \mathrm{COD}$, $O A=O C$ (Given) $O B=O D$ (Given) $\angle A O B=\angle C O D$ (Vertically opposite angles) $\triangle \mathrm{AOB} \cong \triangle \mathrm{C...

## ABCD is a parallelogram. The circle through A, B and C intersect CD (produced if necessary) at E.

[question] Question. ABCD is a parallelogram. The circle through A, B and C intersect CD (produced if necessary) at E. Prove that AE = AD. [/question] [solution] Solution: It can be observed that ABCE is a cyclic quadrilateral and in a cyclic quadrilateral, the sum of the opposite angles is 180°. $\angle A E C+\angle C B A=180^{\circ}$ $\angle \mathrm{AEC}+\angle \mathrm{AED}=180^{\circ}($ Linear pair $)$ $\angle A E D=\angle C B A \ldots(1)$ For a parallelogram, opposite angles are equal. $\ang...

## Prove that the circle drawn with any side of a rhombus as diameter passes

[question] Question. Prove that the circle drawn with any side of a rhombus as diameter passes through the point of intersection of its diagonals. [/question] [solution] Solution: Let ABCD be a rhombus in which diagonals are intersecting at point O and a circle is drawn while taking side CD as its diameter. We know that a diameter subtends 90° on the arc. $\therefore \angle C O D=90^{\circ}$ Also, in rhombus, the diagonals intersect each other at $90^{\circ}$. $\angle A O B=\angle B O C=\angle C...

## The lengths of two parallel chords of a circle are 6 cm and 8 cm.

[question] Question. The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at distance 4 cm from the centre, what is the distance of the other chord from the centre? [/question] [solution] Solution: Let AB and CD be two parallel chords in a circle centered at O. Join OB and OD. Distance of smaller chord AB from the centre of the circle = 4 cm $O M=4 \mathrm{~cm}$ $\mathrm{MB}=\frac{\mathrm{AB}}{2}=\frac{6}{2}=3 \mathrm{~cm}$ In $\triangle O M B$ $\mathrm{OM}^{...

## Two chords AB and CD of lengths 5 cm 11cm respectively of a circle are parallel to each other and are on opposite sides of its centre.

[question] Question. Two chords AB and CD of lengths 5 cm 11cm respectively of a circle are parallel to each other and are on opposite sides of its centre. If the distance between AB and CD is 6 cm, find the radius of the circle. [/question] [solution] Solution: Draw $O M \perp A B$ and $O N \perp C D$. Join $O B$ and $O D$. $\mathrm{BM}=\frac{\mathrm{AB}}{2}=\frac{5}{2}$ (Perpendicular from the centre bisects the chord) $\mathrm{ND}=\frac{\mathrm{CD}}{2}=\frac{11}{2}$ Let $O N$ be $x$. Therefor...

## Prove that line of centres of two intersecting circles subtends

[question] Question. Prove that line of centres of two intersecting circles subtends equal angles at the two points of intersection. [/question] [solution] Solution: Let two circles having their centres as $O$ and $O^{\prime}$ intersect each other at point $A$ and $B$ respectively. Let us join $O O^{\prime}$. In $\triangle \mathrm{AO} \mathrm{O}^{\prime}$ and $\mathrm{BO} \mathrm{O}^{\prime}$, OA $=$ OB (Radius of circle 1) $\mathrm{O}^{\prime} \mathrm{A}=\mathrm{O}^{\prime} \mathrm{B}$ (Radius ...

## Prove that a cyclic parallelogram is a rectangle.

[question] Question. Prove that a cyclic parallelogram is a rectangle. [/question] [solution] Solution: Let ABCD be a cyclic parallelogram. $\angle \mathrm{A}+\angle \mathrm{C}=180^{\circ}$ (Opposite angles of a cyclic quadrilateral) ... (1) We know that opposite angles of a parallelogram are equal. $\therefore \angle A=\angle C$ and $\angle B=\angle D$ From equation (1), $\angle \mathrm{A}+\angle \mathrm{C}=180^{\circ}$ $\Rightarrow \angle \mathrm{A}+\angle \mathrm{A}=180^{\circ}$ $\Rightarrow ...

## ABC and ADC are two right triangles with common hypotenuse AC.

[question] Question. ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠CAD = ∠CBD. [/question] [solution] Solution: In $\triangle \mathrm{ABC}$, $\angle A B C+\angle B C A+\angle C A B=180^{\circ}$ (Angle sum property of a triangle) $\Rightarrow 90^{\circ}+\angle B C A+\angle C A B=180^{\circ}$ $\Rightarrow \angle B C A+\angle C A B=90^{\circ} \ldots(1)$ In $\triangle \mathrm{ADC}$ $\angle C D A+\angle A C D+\angle D A C=180^{\circ}$ (Angle sum property of a triangle) $\...

## If circles are drawn taking two sides of a triangle as diameters,

[question] Question. If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side. [/question] [solution] Solution: Consider a $\triangle \mathrm{ABC}$. Two circles are drawn while taking $A B$ and $A C$ as the diameter. Let they intersect each other at $D$ and let $D$ not lie on $B C$. Join AD. $\angle A D B=90^{\circ}$ (Angle subtended by semi-circle) $\angle \mathrm{ADC}=90^{\circ}$ (Angle subtended by semi-circl...

## Two circles intersect at two points B and C.

[question] Question. Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see the given figure). Prove that ∠ACP = ∠QCD. [/question] [solution] Solution: Join chords AP and DQ. For chord AP, $\angle P B A=\angle A C P$ (Angles in the same segment)...(1) For chord DQ, $\angle D B Q=\angle Q C D$ (Angles in the same segment) ... (2) $A B D$ and $P B Q$ are line segments intersecting at $B$. $\therefo...

## If the non-parallel sides of a trapezium are equal,

[question] Question. If the non-parallel sides of a trapezium are equal, prove that it is cyclic. [solution] Solution: Consider a trapezium $A B C D$ with $A B|| C D$ and $B C=A D$. Draw AM $\perp C D$ and BN $\perp C D$. In $\triangle \mathrm{AMD}$ and $\triangle \mathrm{BNC}$, $A D=B C$ (Given) $\angle \mathrm{AMD}=\angle \mathrm{BNC}\left(\right.$ By construction, each is $\left.90^{\circ}\right)$ $\mathrm{AM}=\mathrm{BN}$ (Perpendicular distance between two parallel lines is same) $\therefor...

## If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral,

[question] Question. If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle. [/question] [solution] Solution: Let $A B C D$ be a cyclic quadrilateral having diagonals $B D$ and $A C$, intersecting each other at point $O$. $\angle B A D=\frac{1}{2} \angle B O D=\frac{180^{\circ}}{2}=90^{\circ}$ (Consider BD as a chord) $\angle B C D+\angle B A D=180^{\circ}($ Cyclic quadrilateral $)$ $\angle B C D=180^{\circ}-90^{...

## ABCD is a cyclic quadrilateral whose diagonals intersect at a point E

[question] Question. $A B C D$ is a cyclic quadrilateral whose diagonals intersect at a point $E$. If $\angle D B C=70^{\circ}, \angle B A C$ is $30^{\circ}$, find $\angle B C D$. Further, if $A B=B C$, find $\angle E C D$. [/question] [solution] Solution: For chord CD, $\angle C B D=\angle C A D$ (Angles in the same segment) $\angle C A D=70^{\circ}$ $\angle B A D=\angle B A C+\angle C A D=30^{\circ}+70^{\circ}=100^{\circ}$ $\angle B C D+\angle B A D=180^{\circ}$ (Opposite angles of a cyclic qu...

## In the given figure, A, B, C and D are four points on a circle.

[question] Question. In the given figure, $A, B, C$ and $D$ are four points on a circle. $A C$ and $B D$ intersect at a point $E$ such that $\angle B E C=130^{\circ}$ and $\angle E C D=20^{\circ}$. Find $\angle B A C$. [/question] [solution] Solution: In $\triangle C D E$, $\angle C D E+\angle D C E=\angle C E B$ (Exterior angle) $\Rightarrow \angle C D E+20^{\circ}=130^{\circ}$ $\Rightarrow \angle C D E=110^{\circ}$ However, $\angle B A C=\angle C D E$ (Angles in the same segment of a circle) $...

## In the given figure, ∠ABC = 69°,

[question] Question. In the given figure, $\angle A B C=69^{\circ}, \angle A C B=31^{\circ}$, find $\angle B D C$. [/question] [solution] Solution: In $\triangle A B C$ $\angle B A C+\angle A B C+\angle A C B=180^{\circ}$ (Angle sum property of a triangle) $\Rightarrow \angle B A C+69^{\circ}+31^{\circ}=180^{\circ}$ $\Rightarrow \angle B A C=180^{\circ}-100^{\circ}$ $\Rightarrow \angle B A C=80^{\circ}$ $\angle B D C=\angle B A C=80^{\circ}$ (Angles in the same segment of a circle are equal) [/s...