Class 10 ICSE Maths Circles Important Notes

As we know that a circle is a closed two-dimensional geometrical figure, such that all points on the surface of a circle are equidistant from the point called the “centre”. The distance from the centre to any point on the surface of a circle is called “Radius”.

Class 10 ICSE Circles Important Notes

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Board Questions
Important Questions





Circles Important Notes

Q.1 In the adjoining figure, O is the centre of the circle and ∠AOC = 60°. Prove that 3∠y - 2∠x = 140°

Solution

Q2 In the adjoining figure, A, B, C and D are four points on a circle. AC and BD intersect at a point E. If ∠BEC = 130° and ∠ECD = 20°, find ∠BAC

Solution

Q3 In the given figure, ∠BAD = 65°, ∠ABD = 70° and ∠BDC = 45°. (i) Prove that AC is a diameter of the circle. (ii) Find ∠ACB

Solution

Q4 In the adjoining figure, ED is a chord parallel to the diameter AC of the circle ABCDE. If ∠CBE = 63°, calculate ∠DEC

Solution

Q5 In the adjoining figure, AD is a diameter of a circle with centre 0. If AD is parallel to BC and ∠CBD = 32°, find (i) ∠OBD (ii) ∠AOB (iii) ∠BED

Solution

Q6 In the given figure, PQ is a diameter of the circle whose centre is O. Given ∠ROS = 42°, calculate ∠RTS

Solution

Q7 In the adjoining figure, C and D are points on the circumference of the semicircle described on AB as diameter. Given ∠BAD = 70° and ∠DBC = 30°, calculate ∠ABD and ∠BDC

Solution

Q8 In the adjoining figure, ∠BQR = 45° and x = 2y. Calculate the values of x and y

Solution

Q9 In the given figure, O is the centre of the circle. If ∠DAE = 70° , find (giving suitable reasons) the measures of: (i) ∠BCD, (ii) ∠BOD, (iii) ∠OBD

Solution

Q10 In a quadrilateral ABCD, if AB = 6 cm, AD = 8 cm, diagonal BD = 10 cm and ∠BCD = 90°, prove that ∠DAC = ∠DBC. Also compute the perimeter of the quad. ABCD, given that CD = 5 √3cm

Solution

Q11 In the given figure, PQRS is a cyclic quadrilateral. PQ and SR produced meet at T. (i) Prove that ∆TPS ~ ∆TRQ. (ii) Find SP if TP = 18 cm, RQ = 4 cm and TR = 6 cm. (iii) Find the area of quadrilateral PQRS if area of ∆PTS = 27 cm2.

Solution

Q12 There are two concentric circles of radii 3 cm and 5 cm respectively. Find the length of the chord of the outer circle which touches the inner circle

Solution

Q13 In the adjoining figure, two circles touch each other externally at C. Prove that the common tangent at C bisects the other two common tangents

Solution

Q14 In the adjoining figure, a circle is inscribed in the quadrilateral ABCD. Given that BC = 38 cm, QB = 27 cm and DC = 25 cm and that AD is perpendicular to DC, find the radius of the circle

Solution

Q15 In the given figure, AB is a diameter of a circle with centre O and AT is a tangent. If ∠AOQ = 58°, find ∠ATQ

Solution

Q16 In the adjoining figure, PQL and PRM are tangents to the circle with centre O at the points Q and R, respectively and S is a point on the circle such that ∠SQL = 50° and ∠SRM = 60°. Find ∠QSR

Solution

Q17 In the figure , O is the centre of the circle. Determine ∠APB, if PA and PB are tangents and ∠APB = 50°

Solution

Q18 In the given circle with centre O, ∠ABC = 100°, ∠ACD = 40° and CT is a tangent to the circle at C. Find ∠ADC and ∠DCT

Solution

Q19 In the adjoining figure, AB is a diameter of a circle with centre O and CP is the tangent to the circle at the point C. (i) If AP = 20 cm and CP = 10 cm, find the radius of the circle. (ii) If CP = 15 cm and BP = 7.5 cm, find the radius of the circle

Solution

Q20 In the given figure, PT touches whose centre is O at R. Diameter SQ when meets PT at P. If ∠SPR = x° and ∠QRP = y°, show that : x° + 2y° = 90°

Solution

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