QQuestion
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.
ICSEClass XMathematicsCircles04 Mark2026 Moderate
✓Answer
Since AB is a diameter, the angle in a semicircle is a right angle.
Therefore, ∠ACB = 90°.
By the tangent-chord theorem:
∠DBC = ∠BAC = x
and ∠ABE = ∠BCA = y
Angles on the straight line DBE give:
x + 90° + y = 180°
⇒ x + y = 90° …(1)
In triangle ADB:
∠ADB + ∠DAB + ∠ABD = 180°
⇒ 32° + x + (x + 90°) = 180°
⇒ 2x + 122° = 180°
⇒ 2x = 58°
⇒ x = 29°
Substituting x = 29° in (1):
29° + y = 90°
⇒ y = 61°
Therefore, x = 29° and y = 61°.
Was this answer helpful?
Related Questions
More ICSE Moderate level questions
1/12
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.
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.
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, 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, 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.
In the given diagram, an isosceles △ABC is inscribed in a circle with centre O. PQ is a tangent to the circle at C. OM is perpendicular to chord AC and ∠COM = 65°. Find: (a) ∠ABC (b) ∠BAC (c) ∠BCQ
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 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