QQuestion
In the given figure ‘O’ is the centre of the circle. PQ is a tangent to the circle at B and AB = AC. If ∠CBQ = 40°, find the unknown angles x, y, z and w.
ICSEClass XMathematicsCircles04 Mark2026 Easy
✓Answer
Given, ∠CBQ = 40°
In a circle, the angle between a tangent and a chord through the point of contact is equal to the angle in the opposite (alternate) segment of the circle.
∠BAC = ∠CBQ = 40°
x = 40°
Since, AB = AC.
∠ABC = ∠BCA [Angles opposite to equal sides of a triangle are equal]
In triangle ABC,
∠ABC + ∠BAC + ∠BCA = 180°
2∠ABC + 40° = 180°
2∠ABC = 180° – 40°
2∠ABC = 140°
∠ABC = 70°.
We know that,
The angle which an arc subtends at the center is double that which it subtends at any point on the remaining part of the circumference.
∠BOC = 2∠BAC
y = 2x
y = 80°.
In triangle OBC,
OB = OC (Radii of same circle)
∠OBC = ∠OCB (Angles opposite to equal sides in a triangle are equal)
By angle sum property of triangle,
∠OBC + ∠OCB + ∠BOC = 180°
2∠OBC + 80° = 180°
2∠OBC = 100°
∠OBC = 50°
From figure,
w = ∠ABC – ∠OBC = 70° – 50° = 20°.
We know that,
Sum of opposite angles of a cyclic quadrilateral is 180°.
In cyclic quadrilateral ABCD,
∠ABC + ∠ADC = 180°
70° + z = 180°
z = 180° – 70° = 110°.
Hence, x = 40°, y = 80°, z = 110°, w = 20°.
Was this answer helpful?
Related Questions
More ICSE Easy level questions
1/12
In the given figure, O is the centre of the circle. PQ is a tangent to the circle at T. Chord AB produced meets the tangent at P. AB = 9 cm, BP = 16 cm, ∠PTB = 50° and ∠OBA = 45°. Find: (a) the length of PT (b) ∠BAT (c) ∠BOT (d) ∠ABT
In the given figure, O is the centre of the circle. CE is a tangent to the circle at A. If ∠ABD = 26°, find: (a) ∠BDA (b) ∠BAD (c) ∠CAD (d) ∠ODB
Use ruler and compass for the following construction: (a) construct an equilateral triangle ABC of side 5 cm. (b) construct the circumcircle of ΔABC. (c) construct the locus of points which are equidistant from AB and BC. Mark the point where the circumcircle and locus meet, as D. (d) give the geometrical name of quadrilateral ABCD.
In the given diagram, O is the centre of the circle and the tangent DE touches the circle at B. If ∠ADB = 32°. Find the values of x and y.
In the given figure, ABCD is a cyclic quadrilateral. DE is produced to E and ∠CDE = 65°. If x is the angle subtended by chord AC at the centre, then x is: (a) 32.5° (b) 65° (c) 115° (d) 130°
In the given diagram, O is the centre of the circle. Chord SR produced meets the tangent XTP at P. (a) Prove that △PTR ∼ △PST. (b) Prove that PT² = PR × PS. (c) If PR = 4 cm and PS = 16 cm, find the length of tangent PT.
X, Y, Z and C are points on the circumference of a circle with centre O. AB is a tangent to the circle at X and ZY = XY. Given ∠OBX = 32° and ∠AXZ = 66°, find: (a) ∠BOX (b) ∠CYX (c) ∠ZYX (d) ∠OXY
In the given diagram, chords AC and BC are equal. If ∠ACD = 120°, then ∠AEC is: (a) 30° (b) 60° (c) 90° (d) 120°
In the given diagram, RT is a tangent touching the circle at S. If ∠PST = 30° and ∠SPQ = 60°, then ∠PSQ is equal to: (a) 40° (b) 30° (c) 60° (d) 90°
In the given diagram, O is the centre of the circle. PR and PT are two tangents drawn from the external point P and touching the circle at Q and S respectively. MN is a diameter of the circle. Given ∠PQM = 42° and ∠PSM = 25°. Find: (a) ∠OQM (b) ∠QNS (c) ∠QOS (d) ∠QMS
In the given diagram, PS and PT are tangents to the circle. SQ ∥ PT and ∠SPT = 80°. The value of ∠QST is: (a) 140° (b) 90° (c) 80° (d) 50°
The figure shows a circle of radius 9 cm with O as the centre. The diameter AB produced meets the tangent PQ at P. If PA = 24 cm, find the length of tangent PQ.