## ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

Question 1.

If O is the centre of the circle, find the value of x in each of the following figures (using the given information):

Solution:

Question 2.

(a) In the figure (i) given below, O is the centre of the circle. If ∠AOC = 150°, find (i) ∠ABC (ii) ∠ADC.

(b) In the figure (ii) given below, AC is a diameter of the given circle and ∠BCD = 75°. Calculate the size of (i) ∠ABC (ii) ∠EAF.

Solution:

Question 3.

(a) In the figure, (i) given below, if ∠DBC = 58° and BD is a diameter of the circle, calculate:

(i) ∠BDC (ii) ∠BEC (iii) ∠BAC

(b) In the figure (if) given below, AB is parallel to DC, ∠BCE = 80° and ∠BAC = 25°. Find:

(i) ∠CAD (ii) ∠CBD (iii) ∠ADC (2008)

Solution:

Question 4.

(a) In the figure given below, ABCD is a cyclic quadrilateral. If ∠ADC = 80° and ∠ACD = 52°, find the values of ∠ABC and ∠CBD.

(b) In the figure given below, O is the centre of the circle. ∠AOE =150°, ∠DAO = 51°. Calculate the sizes of ∠BEC and ∠EBC.

Solution:

Question 5.

(a) In the figure (i) given below, ABCD is a parallelogram. A circle passes through A and D and cuts AB at E and DC at F. Given that ∠BEF = 80°, find ∠ABC.

(b) In the figure (ii) given below, ABCD is a cyclic trapezium in which AD is parallel to BC and ∠B = 70°, find:

(i)∠BAD (ii) DBCD.

Solution:

Question 6.

(a) In the figure given below, O is the centre of the circle. If ∠BAD = 30°, find the values of p, q and r.

(b) In the figure given below, two circles intersect at points P and Q. If ∠A = 80° and ∠D = 84°, calculate

(i) ∠QBC (ii) ∠BCP

Solution:

Question 7.

(a) In the figure given below, PQ is a diameter. Chord SR is parallel to PQ.Given ∠PQR = 58°, calculate (i) ∠RPQ (ii) ∠STP

(T is a point on the minor arc SP)

(b) In the figure given below, if ∠ACE = 43° and ∠CAF = 62°, find the values of a, b and c (2007)

Solution:

Question 8.

(a) In the figure (i) given below, AB is a diameter of the circle. If ∠ADC = 120°, find ∠CAB.

(b) In the figure (ii) given below, sides AB and DC of a cyclic quadrilateral ABCD are produced to meet at E, the sides AD and BC are produced to meet at F. If x : y : z = 3 : 4 : 5, find the values of x, y and z.

Solution:

Question 9.

(a) In the figure (i) given below, ABCD is a quadrilateral inscribed in a circle with centre O. CD is produced to E. If ∠ADE = 70° and ∠OBA = 45°, calculate

(i) ∠OCA (ii) ∠BAC

(b) In figure (ii) given below, ABF is a straight line and BE || DC. If ∠DAB = 92° and ∠EBF = 20°, find :

(i) ∠BCD (ii) ∠ADC.

Solution:

Question 10.

(a) In the figure (ii) given below, PQRS is a cyclic quadrilateral in which PQ = QR and RS is produced to T. If ∠QPR = 52°, calculate ∠PST.

(b) In the figure (ii) given below, O is the centre of the circle. If ∠OAD = 50°, find the values of x and y.

Solution:

Question 11.

(a) In the figure (i) given below, O is the centre of the circle. If ∠COD = 40° and ∠CBE = 100°, then find :

(i) ∠ADC

(ii) ∠DAC

(iii) ∠ODA

(iv) ∠OCA.

(b) In the figure (ii) given below, O is the centre of the circle. If ∠BAD = 75° and BC = CD, find :

(i) ∠BOD

(ii) ∠BCD

(iii) ∠BOC

(iv) ∠OBD (2009)

Solution:

Question 12.

In the given figure, O is the centre and AOE is the diameter of the semicircle ABCDE. If AB = BC and ∠AEC = 50°, find :

(i) ∠CBE

(ii) ∠CDE

(iii) ∠AOB.

Prove that OB is parallel to EC.

Solution:

Question 13.

(a) In the figure (i) given below, ED and BC are two parallel chords of the circle and ABE, ACD are two st. lines. Prove that AED is an isosceles triangle.

(b) In the figure (ii) given below, SP is the bisector of ∠RPT and PQRS is a cyclic quadrilateral. Prove that SQ = RS.

Solution:

Question 14.

In the given figure, ABC is an isosceles triangle in which AB = AC and circle passing through B and C intersect sides AB and AC at points D and E. Prove that DE || BC.

Solution:

Question 15.

(a) Prove that a cyclic parallelogram is a rectangle.

(b) Prove that a cyclic rhombus is a square.

Solution:

Question 16.

In the given figure, chords AB and CD of the circle are produced to meet at O. Prove that triangles ODB and OAC are similar. Given that CD = 2 cm, DO = 6 cm and BO = 3 cm, area of quad. CABD

Solution: