Mathematics
In the given figure, O is the center of the circle. The tangents at B and D intersect each other at point P. If AB is parallel to CD and ∠ABC = 55°, find :
(i) ∠BOD
(ii) ∠BPD
Circles
1 Like
Answer
(i) From figure,
∠BCD = ∠ABC = 55° [Alternate angles are equal.]
We know that,
The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
∴ ∠BOD = 2∠BCD = 2 x 55° = 110°.
Hence, ∠BOD = 110°.
(ii) We know that,
A tangent line is always at a right angle to the radius of the circle at the point of tangency.
∴ ∠OBP = 90° and ∠ODP = 90°.
In quadrilateral ODPB,
⇒ ∠BOD + ∠OBP + ∠ODP + ∠BPD = 360° [Angle sum property of quadrilateral]
⇒ 110° + 90° + 90° + ∠BPD = 360°
⇒ ∠BPD = 360° - 290° = 70°.
Hence, ∠BPD = 70°.
Answered By
1 Like
Related Questions
In the following figure, PQ = QR, ∠RQP = 68°, PC and CQ are tangents to the circle with center O.
Calculate the values of :
(i) ∠QOP
(ii) ∠QCP
In the figure (i) given below, AB is a diameter. The tangent at C meets AB produced at Q, ∠CAB = 34°. Find :
(i) ∠CBA
(ii) ∠CQA
In the given figure, QAP is the tangent at point A and PBD is a straight line.
If ∠ACB = 36° and ∠APB = 42°, find :
(i) ∠BAP
(ii) ∠ABD
(iii) ∠QAD
(iv) ∠BCD
In the figure, given below, AC is a transverse common tangent to two circles with centers P and Q and of radii 6 cm and 3 cm respectively. Given that AB = 8 cm, calculate PQ.
