Frank solutions for Mathematics - Part 2 [English] Class 10 ICSE chapter 17 - Circles [Latest edition]

Find the length of the chord of a circle in the following when: 

Radius is 13 cm and the distance from the centre is 12 cm 

Find the length of the chord of a circle in the following when: 

Radius is 1. 7cm and the distance from the centre is 1.5 cm 

Find the length of the chord of a circle in the following when: 

Radius is 6.5 cm and the distance from the centre is 2.5 cm 

AB and CD are two equal chords of a drde intersecting at Pas shown in fig. P is joined to O , the centre of the cirde. Prove that OP bisects  ∠ CPB. 

In fig. the centre of the circle is O. PQ and RS are two equal chords of the circle which , when produced , meet at T outside the circle . Prove that (a) TP = TR (b) TQ = TS.

PQ and QR are two equal chords of a circle. A diameter of the circle is drawn through Q . Prove that the diameter bisects ∠ PQR.

M and N are the midpoints of chords AB and CD . The line MN passes through the centre O . Prove that AB || CD.

Prove that the line segment joining the midpoints of two parallel chords of a circle passes through its centre. 

Two congruent drdes have their centres at 0 and P. Mis the midpoint of the line segment OP. A straight line is drawn through M cutting the two circles at the points A, B, C and D. Prove that the chords AB and CD are equal. 

Two circles with centres O and P intersect each other at A and B as shown in following fig. Two straight lines MAN and RBQ are drawn parallel to OP.
Prove that (i) MN = 20 P (ii) MN= RQ.

In following fig. ABC is an equilateral triangle . A circle is drawn with centre A so that ot cuts AB and AC at M and N respectively. Prove that BN = CM.

In following fig., ABCD is a square. A cirde is drawn with centre A so that it cuts AB and AD at Mand N respectively. Prove that  Δ DAM  ≅  Δ .BAN. 

In fig, AB and CD are two equal chords of a circle with centre O. If M and N are the midpoints of AB and CD respectively,

prove that (a) ∠ ONM = ∠ ONM (b) ∠ AMN = ∠ CNM. 

Find the diameter of the circle if the length of a chord is 3.2 cm and itd distance from the centre is 1.2 cm.

In following figure .,XY and YZ are two equal chords of a circle with centre O. Prove that the bisector of ∠ XYZ passes through O.

In figure , ABC is an isosceles triangle inscribed in a circle with centre O such that AB = AC = 13 cm and BC = 10 cm .Find the radius of the circle.

AB and AC are two equal chords of a circle with centre o such that LABO and LCBO are equal. Prove that AB = BC. 

A chord of a length 16.8 cm is at a distance of 11.2 cm from the centre of a circle . Find the length of the chord of the same circle which is at a distance of 8.4 cm from the centre.

A chord of length 6 cm is at a distance of 7.2 cm from the centre of a circle. Another chord of the same circle is of length 14.4 cm. Find its distance from the centre.

A chord of length 8cm is drawn inside a circle of radius 6cm. Find the perpendicular distance of the chord from the centre of the circle.

Two circles of radii 5cm and 3cm with centres O and P touch each other internally. If the perpendicular bisector of the line segment OP meets the circumference of the larger circle at A and B, find the length of AB.

Two chords AB and CD of lengths 6cm and 12cm are drawn parallel inside the circle. If the distance between the chords of the circle is 3cm, find the radius of the circle.

Two chords of lengths 10cm and 24cm are drawn parallel o each other in a circle. If they are on the same side of the centre and the distance between them is 17cm, find the radius of the circle.

In following figure , AB , a chord of the circle is of length 18 cm. It is perpendicularly bisected at M by PQ. 

In following figure , O is the centre of the circle. If ∠ APB = 50° then find ∠ AOB and ∠ OAB.

In a cyclic quadrilateral ABCD , AB || CD and ∠ B = 65 ° , find the remaining angles

In a cyclic quadrilateral ABCD , AB || CD and ∠ B = 65° , find the remaining angles.

In a cyclic quadrialteral ABCD , if m ∠ A = 3 (m ∠C). Find m ∠ A.

In a circle with centre O , chords AB and CD intersets inside the circle at E . Prove that ∠ AOC = ∠ BOD = 2 ∠ AEC.

In the given Figure, AB and CD are two chords of a circle, intersecting each other at P such that AP = CP. Show that AB= CD. 

MABN are points on a drde having centre O. AN and MB cut at Y. If ∠ NYB = 50" and ∠ YNB = 200, find ∠  MAN and reflex angle MON.

In fig., PT is a tangent to the circle at T and PAB is a secant to the same circle. If PB = 9cm and AB = 5 cm, find PT.

Two circles are drawn with sides AB, AC of a triangle ABC as diameters. They intersect at a point D. Prove that D lies on BC.

Prove that the circle drawn on any one of the equal sides of an isosceles triangle as diameter bisects the base. 

Prove that the circle drawn with any side of a rhombus as a diameter, passes through the point of intersection of its diagonals. 

In fig., O is the centre of the circle and ∠ AOC = 1500. Find ∠ ABC. 

In triangle ABC, AB = AC. A circle passing through B and c intersects the sides AB and AC at D and E respectively. Prove that DE || BC. 

The bisectors of the opposite angles A and C of a cydic quadrilateral ABCD intersect the cirde at the points E and F, respectively. Prove that EF is a diameter of the circle.

Prove that the angles bisectors of the angles formed by producing opposite sides of a cyclic quadrilateral (provided they are not parallel) intersect at right triangle.

Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively. Prove that the angles of  Δ DEF are 90° - `"A"/2` , 90° - `"B"/2` and 90° - `"C"/2` respectively.

In following figure . O is the centre of the circle. Find ∠ BAC.

In following  fig., O is the centre of the circle, prove that ∠x =∠ y + ∠ z. 

In following figure , chord ED is parallel to the diameter AC of the circle. Given ∠ CBE = 65° , calculate ∠DEC .

In following figure , C is a point on the minor arc AB of the circle with centre O . Given ∠ ACB = p° , ∠  AOB = q° , express q in terms of p. Calculate p if OACB is a parallelogram.

In following figure , Δ PQR is an isosceles teiangle with PQ = PR and m ∠ PQR = 35° .Find m ∠ QSR and ∠ QTR 

In following figure.,ABCD is a cyclic quadrilateral . If ∠ BCD = 100° and ∠ ABD = 70° , find ∠ ADB.

In following fig., O is the centre of the circle. Find ∠ CBD. 

Find the length of the tangent from a point which is at a distance of 5cm from the centre of the circle of radius 3cm.

From a point P outside a circle, with centre O. tangents PA and PB are drawn as following fig., Prove that ∠ AOP = ∠ BOP and OP is the perpendicular bisector of AB. 

In fig., chords AB and CD of a circle intersect at P. AP = 5cm, BP= 3cm and CP = 2.5cm. Determine the length of DP.

In following fig., chords PQ and RS of a circle intersect at T. If RS = 18cm, ST = 6cm and PT = 18cm, find the length of TQ. 

In fig., AB and DC are two chords of a circle with centre O. these chords when produced meet at P. if PB = Bern, BA = 7cm and PO = 14.5cm, find the radius of the circle. 

In following fig., PT is a tangent to the circle at T and PAB is a secant to the same circle. If PA = 4cm and AB = Scm, find PT.

In following fig., PT is a tangent to the circle at T and PAB is a secant to the same circle. If PA = 4cm and AB = Scm, find PT.

In following fig., PT is a tangent to the circle at T and PAB is a secant to the same circle. If PB = 9cm and AB = Scm, find PT. 

In following fig., PT is tangent to the circle at T and CD is a diameter of the same circle. If PC= 3cm and PT= 6cm, find the radius of the circle.

The length of the direct common tangent to two circles of radii 12cm and 4cm is 15cm. calculate the distance between their centres.

Calculate the length of direct common tangent to two circles of radii 3cm and Bern with their centres 13cm apart.

A point A is 17cm from the centre of the circle. The length of the tangent drawn from A to the circle is 15cm. find the radius of the circle.

In the following  fig. , AC is a transversal common tangent to tvvo circles with centres P and Q and of radii 6cm and 3cm respectively. Given that AB = 8cm, calculate PQ. 

In following fig., ABC is a right- angled triangle at A with sides AB = 5 cm and BC = 13 cm . A circle with centre O has been inscribed in the triangle ABC. Calculate the radius of the incircle. 

In followinf fig., two concentric circles with centre 0 are of radii 5 cm and 3 cm. from an external point P, tangents PA and PB are drawn to these circles. If AP = 12cm, find BP.

In the figure, XP and XQ are tangents from X to the circle with centre O. R is a point on the circle. Prove that XA + AR = XB + BR. 

PA and PB are tangents from P to the circle with centre O. At M, a tangent is drawn cutting PA at K and PB at N. Prove that KN = AK + BN.

Two tangents are drawn to a circle from an external point P. touching the circle at the points A and B. A third tangent intersects segment PA in C and segment PB in D and touches the circle at Q. if PA = 20 units, find the perimeter of Δ  PCD. 

In following figure , the incircle of Δ ABC , touches the sides BC , CA and AB at D , E and F respectively. Show AF + BD + CE = AE + BF + CD

In following fig., a circle is touching the side BC of Δ ABC at P and AB and AC produced at Q and R respectively. Prove that AQ is half the perimeter of Δ ABC. 

If all the sides of a parallelogram touch a circle, show that the parallelogram is a rhombus.

If Δ PQR is isosceles with PQ = PR and a circle with centre O and radius r is the incircle of the  Δ PQR touching QR at T, prove that the point T bisects QR.