ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 1.
Find the length of the tangent drawn to a circle of radius 3 cm, from a point distant 5 cm from the centre.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

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Question 2.
A point P is at a distance 13 cm from the centre C of a circle and PT is a tangent to the given circle. If PT = 12 cm, find the radius of the circle.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 3.
The tangent to a circle of radius 6 cm from an external point P, is of length 8 cm. Calculate the distance of P from the nearest point of the circle.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 4.
Two concentric circles are of the radii 13 cm and 5 cm. Find the length of the chord of the outer circle which touches the inner circle.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 5.
Two circles of radii 5 cm and 2-8 cm touch each other. Find the distance between their centres if they touch:
(i) externally
(ii) internally.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 6.
(a) In figure (i) given below, triangle ABC is circumscribed, find x.
(b) In figure (ii) given below, quadrilateral ABCD is circumscribed, find x.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 7.
(a) In figure (i) given below, quadrilateral ABCD is circumscribed; find the perimeter of quadrilateral ABCD.
(b) In figure (ii) given below, quadrilateral ABCD is circumscribed and AD ⊥ DC ; find x if radius of incircle is 10 cm.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 8.
(a) In the figure (i) given below, O is the centre of the circle and AB is a tangent at B. If AB = 15 cm and AC = 7.5 cm, find the radius of the circle.
(b) In the figure (ii) given below, from an external point P, tangents PA and PB are drawn to a circle. CE is a tangent to the circle at D. If AP = 15 cm, find the perimeter of the triangle PEC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 9.
(a) If a, b, c are the sides of a right triangle where c is the hypotenuse, prove that the radius r of the circle which touches the sides of the triangle is given by
\(r= \frac { a+b-c }{ 2 } \)
(b) In the given figure, PB is a tangent to a circle with centre O at B. AB is a chord of length 24 cm at a distance of 5 cm from the centre. If the length of the tangent is 20 cm, find the length of OP.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 10.
Three circles of radii 2 cm, 3 cm and 4 cm touch each other externally. Find the perimeter of the triangle obtained on joining the centres of these circles.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 11.
(a) In the figure (i) given below, the sides of the quadrilateral touch the circle. Prove that AB + CD = BC + DA.
(b) In the figure (ii) given below, ABC is triangle with AB = 10cm, BC = 8cm and AC = 6cm (not drawn to scale). Three circles are drawn touching each other with vertices A, B and C as their centres. Find the radii of the three circles
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 12.
(a) ln the figure (i) PQ = 24 cm, QR = 7 cm and ∠PQR = 90°. Find the radius of the inscribed circle ∆PQR
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
(b) In the figure (ii) given below, two concentric circles with centre O 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.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 13.
(a) In the figure (i) given below, AB = 8 cm and M is mid-point of AB. Semi-circles are drawn on AB, AM and MB as diameters. A circle with centre C touches all three semi-circles as shown, find its radius.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
(b) In the figure (ii) given below, equal circles with centres O and O’ touch each other at X. OO’ is produced to meet a circle O’ at A. AC is tangent to the circle whose centre is O. O’D is perpendicular to AC. Find the value of :
(i) \(\\ \frac { AO’ }{ AO } \)
(ii) \(\frac { area\quad of\quad \Delta ADO’ }{ area\quad of\quad \Delta ACO } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 14.
The length of the direct common tangent to two circles of radii 12 cm and 4 cm is 15 cm. Calculate the distance between their centres.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 15.
Calculate the length of a direct common tangent to two circles of radii 3 cm and 8 cm with their centres 13 cm apart.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 16.
In the given figure, AC is a transverse common tangent to two circles with centres P and Q and of radii 6 cm and 3 cm respectively. Given that AB = 8 cm, calculate PQ.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 17.
Two circles with centres A, B are of radii 6 cm and 3 cm respectively. If AB = 15 cm, find the length of a transverse common tangent to these circles.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 18.
(a) In the figure (i) given below, PA and PB are tangents at a points A and B respectively of a circle with centre O. Q and R are points on the circle. If ∠APB = 70°, find (i) ∠AOB (ii) ∠AQB (iii) ∠ARB
(b) In the figure (ii) given below, two circles touch internally at P from an external point Q on the common tangent at P, two tangents QA and QB are drawn to the two circles. Prove that QA = QB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 19.
In the given figure, AD is a diameter of a circle with centre O and AB is tangent at A. C is a point on the circle such that DC produced intersects the tangent af B. If ∠ABC = 50°, find ∠AOC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 20.
In the given figure, tangents PQ and PR are drawn from an external point P to a circle such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ, Find ∠RQS
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 21.
(a) In the figure (i) given below, PQ is a tangent to the circle at A, DB is a diameter, ∠ADB = 30° and ∠CBD = 60°, calculate (i) ∠QAB (ii) ∠PAD (iii) ∠CDB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
(b) In the figure (ii) given below, ABCD is a cyclic quadrilateral. The tangent to the circle at B meets DC produced at F. If ∠EAB = 85° and ∠BFC = 50°, find ∠CAB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 22.
(a) In the figure (i) given below, O is the centre of the circle and SP is a tangent. If ∠SRT = 65°, find the value of x, y and z. (2015)
(b) In the figure (ii) given below, O is the centre of the circle. PS and PT are tangents and ∠SPT = 84°. Calculate the sizes of the angles TOS and TQS.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 23.
In the given figure, O is the centre of the circle. Tangents to the circle at A and B meet at C. If ∠ACO = 30°, find
(i) ∠BCO (ii) ∠AOR (iii) ∠APB
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 24.
(a) In the figure (i) given below, O is the centre of the circle. The tangent at B and D meet at P. If AB is parallel to CD and ∠ ABC = 55°. find (i) ∠BOD (ii) ∠BPD
(b) In the figure (ii) given below. O is the centre of the circle. AB is a diameter, TPT’ is a tangent to the circle at P. If ∠BPT’ = 30°, calculate : (i) ∠APT (ii) ∠BOP.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 25.
In the adjoining figure, ABCD is a cyclic quadrilateral.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
The line PQ is the tangent to the circle at A. If ∠CAQ : ∠CAP = 1 : 2, AB bisects ∠CAQ and AD bisects ∠CAP, then find the measure of the angles of the cyclic quadrilateral. Also prove that BD is a diameter of the circle.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 26.
In a triangle, ABC, the incircle (centre O) touches BC, CA and AB at P, Q and R respectively. Calculate (i) ∠QOR (ii) ∠QPR given that ∠A = 60°.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 27.
(a) In figure (0 given below, AB is a diameter. The tangent at C meets AB produced at Q, ∠CAB = 34°. Find
(i) ∠CBA (ii) ∠CQA (2006)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
(b) In the figure (ii) given below, AP and BP are tangents to the circle with centre O. Given ∠APB = 60°, calculate.
(i) ∠AOB (ii) ∠OAB (iii) ∠ACB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 28.
(a) In the figure (i) given below, O is the centre of the circumcircle of triangle XYZ. Tangents at X and Y intersect at T. Given ∠XTY = 80° and ∠XOZ = 140°, calculate the value of ∠ZXY. (1994)
(b) In the figure (ii) given below, O is the centre of the circle and PT is the tangent to the circle at P. Given ∠QPT = 30°, calculate (i) ∠PRQ (ii) ∠POQ.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 29.
Two chords AB, CD of a circle intersect internally at a point P. If
(i) AP = cm, PB = 4 cm and PD = 3 cm, find PC.
(ii) AB = 12 cm, AP = 2 cm, PC = 5 cm, find PD.
(iii) AP = 5 cm, PB = 6 cm and CD = 13 cm, find CP.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 30.
(a) In the figure (i) given below, PT is a tangent to the circle. Find TP if AT = 16 cm and AB = 12 cm.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
(b) In the figure given below, diameter AB and chord CD of a circle meet at P. PT is a tangent to the circle at T. CD = 7.8 cm, PD = 5 cm, PB = 4 cm. Find (i) AB. (ii) the length of tangent PT.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 31.
PAB is secant and PT is tangent to a circle
(i) PT = 8 cm and PA = 5 cm, find the length of AB.
(ii) PA = 4.5 cm and AB = 13.5 cm, find the length of PT.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 32.
In the adjoining figure, CBA is secant and CD is tangent to the circle. If AB = 7 cm and BC = 9 cm, then
(i) Prove that ∆ACD ~ ∆DCB.
(ii) Find the length of CD.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 33.
(a) In the figure (i) given below, PAB is secant and PT is tangent to a circle. If PA : AB = 1 : 3 and PT = 6 cm, find the length of PB.
(b) In the figure (ii) given below, ABC is an isosceles triangle in which AB = AC and Q is mid-point of AC. If APB is a secant, and AC is tangent to the circle at Q, prove that AB = 4 AP.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 34.
Two chords AB, CD of a circle intersect externally at a point P. If PA = PC, Prove that AB = CD.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 35.
(a) In the figure (i) given below, AT is tangent to a circle at A. If ∠BAT = 45° and ∠BAC = 65°, find ∠ABC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
(b) In the figure (ii) given below, A, B and C are three points on a circle. The tangent at C meets BA produced at T. Given that ∠ATC = 36° and ∠ACT = 48°, calculate the angle subtended by AB at the centre of the circle. (2001)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 36.
In the adjoining figure, ∆ABC is isosceles with AB = AC. Prove that the tangent at A to the circumcircle of ∆ABC is parallel to BC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 37.
If the sides of a rectangle touch a circle, prove that the rectangle is a square.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 38.
(a) In the figure (i) given below, two circles intersect at A, B. From a point P on one of these circles, two line segments PAC and PBD are drawn, intersecting the other circle at C and D respectively. Prove that CD is parallel to the tangent at P.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
(b) In the figure (ii) given below, two circles with centres C, C’ intersect at A, B and the point C lies on the circle with centre C’. PQ is a tangent to the circle with centre C’ at A. Prove that AC bisects ∠PAB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 39.
(a) In the figure (i) given below, AB is a chord of the circle with centre O, BT is tangent to the circle. If ∠OAB = 32°, find the values of x and y.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
(b) In the figure (ii) given below, O and O’ are centres of two circles touching each other externally at the point P. The common tangent at P meets a direct common tangent AB at M. Prove that:
(i) M bisects AB (ii) ∠APB = 90°.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

ML Aggarwal Class 10 Solutions for ICSE Maths