NCERT Exemplar solutions for Mathematics [English] Class 9 chapter 10 - Circles [Latest edition]

AD is a diameter of a circle and AB is a chord. If AD = 34 cm, AB = 30 cm, the distance of AB from the centre of the circle is ______.

In the following figure, if OA = 5 cm, AB = 8 cm and OD is perpendicular to AB, then CD is equal to ______.

If AB = 12 cm, BC = 16 cm and AB is perpendicular to BC, then the radius of the circle passing through the points A, B and C is ______.

In the following figure, if ∠ABC = 20º, then ∠AOC is equal to ______.

In the following figure, if AOB is a diameter of the circle and AC = BC, then ∠CAB is equal to ______.

In the following figure, if ∠OAB = 40º, then ∠ACB is equal to ______.

In the following figure, if ∠DAB = 60º, ∠ABD = 50º, then ∠ACB is equal to ______.

ABCD is a cyclic quadrilateral such that AB is a diameter of the circle circumscribing it and ∠ADC = 140º, then ∠BAC is equal to ______.

In the following figure, BC is a diameter of the circle and ∠BAO = 60º. Then ∠ADC is equal to ______.

In the following figure, ∠AOB = 90º and ∠ABC = 30º, then ∠CAO is equal to ______.

Two chords AB and CD of a circle are each at distances 4 cm from the centre. Then AB = CD.

Two chords AB and AC of a circle with centre O are on the opposite sides of OA. Then ∠OAB = ∠OAC .

Two congruent circles with centres O and O′ intersect at two points A and B. Then ∠AOB = ∠AO′B.

Through three collinear points a circle can be drawn.

A circle of radius 3 cm can be drawn through two points A, B such that AB = 6 cm.

If AOB is a diameter of a circle and C is a point on the circle, then AC² + BC² = AB².

ABCD is a cyclic quadrilateral such that ∠A = 90°, ∠B = 70°, ∠C = 95° and ∠D = 105°.

If A, B, C, D are four points such that ∠BAC = 30° and ∠BDC = 60°, then D is the centre of the circle through A, B and C.

If A, B, C and D are four points such that ∠BAC = 45° and ∠BDC = 45°, then A, B, C, D are concyclic.

In the following figure, if AOB is a diameter and ∠ADC = 120°, then ∠CAB = 30°.

If arcs AXB and CYD of a circle are congruent, find the ratio of AB and CD.

If the perpendicular bisector of a chord AB of a circle PXAQBY intersects the circle at P and Q, prove that arc PXA ≅ arc PYB.

A, B and C are three points on a circle. Prove that the perpendicular bisectors of AB, BC and CA are concurrent.

AB and AC are two equal chords of a circle. Prove that the bisector of the angle BAC passes through the centre of the circle.

If a line segment joining mid-points of two chords of a circle passes through the centre of the circle, prove that the two chords are parallel.

ABCD is such a quadrilateral that A is the centre of the circle passing through B, C and D. Prove that ∠CBD + ∠CDB = `1/2` ∠BAD

O is the circumcentre of the triangle ABC and D is the mid-point of the base BC. Prove that ∠BOD = ∠A.

On a common hypotenuse AB, two right triangles ACB and ADB are situated on opposite sides. Prove that ∠BAC = ∠BDC.

Two chords AB and AC of a circle subtends angles equal to 90º and 150º, respectively at the centre. Find ∠BAC, if AB and AC lie on the opposite sides of the centre.

If BM and CN are the perpendiculars drawn on the sides AC and AB of the triangle ABC, prove that the points B, C, M and N are concyclic.

If a line is drawn parallel to the base of an isosceles triangle to intersect its equal sides, prove that the quadrilateral so formed is cyclic.

If a pair of opposite sides of a cyclic quadrilateral are equal, prove that its diagonals are also equal.

The circumcentre of the triangle ABC is O. Prove that ∠OBC + ∠BAC = 90º.

A chord of a circle is equal to its radius. Find the angle subtended by this chord at a point in major segment.

In the following figure, ∠ADC = 130° and chord BC = chord BE. Find ∠CBE.

In the following figure, ∠ACB = 40º. Find ∠OAB.

A quadrilateral ABCD is inscribed in a circle such that AB is a diameter and ∠ADC = 130º. Find ∠BAC.

Two circles with centres O and O′ intersect at two points A and B. A line PQ is drawn parallel to OO′ through A(or B) intersecting the circles at P and Q. Prove that PQ = 2 OO′.

In the following figure, AOB is a diameter of the circle and C, D, E are any three points on the semi-circle. Find the value of ∠ACD + ∠BED.

In the following figure, ∠OAB = 30º and ∠OCB = 57º. Find ∠BOC and ∠AOC.

If two equal chords of a circle intersect, prove that the parts of one chord are separately equal to the parts of the other chord.

If non-parallel sides of a trapezium are equal, prove that it is cyclic.

If P, Q and R are the mid-points of the sides BC, CA and AB of a triangle and AD is the perpendicular from A on BC, prove that P, Q, R and D are concyclic.

ABCD is a parallelogram. A circle through A, B is so drawn that it intersects AD at P and BC at Q. Prove that P, Q, C and D are concyclic.

Prove that angle bisector of any angle of a triangle and perpendicular bisector of the opposite side if intersect, they will intersect on the circumcircle of the triangle.

If two chords AB and CD of a circle AYDZBWCX intersect at right angles (see figure), prove that arc CXA + arc DZB = arc AYD + arc BWC = semi-circle.

If ABC is an equilateral triangle inscribed in a circle and P be any point on the minor arc BC which does not coincide with B or C, prove that PA is angle bisector of ∠BPC.

In the following figure, AB and CD are two chords of a circle intersecting each other at point E. Prove that ∠AEC = `1/2` (Angle subtended by arc CXA at centre + angle subtended by arc DYB at the centre).

If bisectors of opposite angles of a cyclic quadrilateral ABCD intersect the circle, circumscribing it at the points P and Q, prove that PQ is a diameter of the circle.

A circle has radius `sqrt(2)` cm. It is divided into two segments by a chord of length 2 cm. Prove that the angle subtended by the chord at a point in major segment is 45°.

Two equal chords AB and CD of a circle when produced intersect at a point P. Prove that PB = PD.

AB and AC are two chords of a circle of radius r such that AB = 2AC. If p and q are the distances of AB and AC from the centre, prove that 4q² = p² + 3r².

In the following figure, O is the centre of the circle, ∠BCO = 30°. Find x and y.

In the following figure, O is the centre of the circle, BD = OD and CD ⊥ AB. Find ∠CAB.