Advertisements
Advertisements
Question
In figure, if PQR is the tangent to a circle at Q whose centre is O, AB is a chord parallel to PR and ∠BQR = 70°, then ∠AQB is equal to ______.
Options
20°
40°
35°
45°
Solution
In figure, if PQR is the tangent to a circle at Q whose centre is O, AB is a chord parallel to PR and ∠BQR = 70°, then ∠AQB is equal to 40°.
Explanation:
Given, AB ॥ PR
∴ ∠ABQ = ∠BQR = 70° ...[Alternate angles]
Also, QD is perpendicular to AB and QD bisects AB.
In ∆QDA and ∆QDB,
∠QDA = ∠QDB ...[90° each]
AD = BD
QD = QD ...[Common side]
∆ADQ ≅ ∆BDQ ...[By SAS congruency]
⇒ ∠QAD = ∠QBD [CPCT]...(i)
But ∠QBD = ∠ABQ = 70°
⇒ ∠QAD = 70° ...[From (i)]
Now, in ∆ABQ,
∠A + ∠B + ∠AQB = 180° ...[Angle sum property]
⇒ ∠AQB = 180° – (70° + 70°) = 40°
APPEARS IN
RELATED QUESTIONS
In the figure given below, PQ is a chord of a circle with centre O and PT is a tangent. If ∠QPT = 60°, find ∠PRQ.
In Fig. 1, AOB is a diameter of a circle with centre O and AC is a tangent to the circle at A. If ∠BOC = 130°, the find ∠ACO.
In fig. 5 is a chord AB of a circle, with centre O and radius 10 cm, that subtends a right angle at the centre of the circle. Find the area of the minor segment AQBP. Hence find the area of major segment ALBQA. (use π = 3.14)
In the following fig. , AC is a transversal common tangent to tvvo circles with centres P and Q and of radii 6cm and 3cm respectively. Given that AB = 8cm, calculate PQ.
PA and PB are tangents from P to the circle with centre O. At M, a tangent is drawn cutting PA at K and PB at N. Prove that KN = AK + BN.
In the given figure, 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 AB.
In Fig. TA is a tangent to a circle from the point T and TBC is a secant to the circle. If AD is the bisector of ∠BAC, prove that ΔADT is isosceles.
In figure, if O is the centre of a circle PQ is a chord and the tangent PR at P makes an angle of 50° with PQ, then ∠POQ is equal to ______.
The quadrilateral formed by joining the angle bisectors of a cyclic quadrilateral is a ______
In the given figure, PT is a tangent to the circle at T, chord BA is produced to meet the tangent at P. Perpendicular BC bisects the chord TA at C. If PA = 9 cm and TB = 7 cm, find the lengths of:
- AB
- PT
