1 prove that the intercept of a tangent between a pair of parallel tangents toa circle substended a right angle - Maths - Circles

Question

1 prove that the intercept of a tangent between a pair of
parallel tangents toa circle substended a right angle at the center
of the circle 2 two tangentsTP andTQ are drawn to a circle with
center O, from an external point T prove that
anglePTQ=angleOPQ

Priyanka Kedia · Accepted Answer

(1)

Let CA and EB are the two tangents to the circle such that CA|| EB And ,AB intersect them at A and B respectively such that AB subtends angle AOB at the centre So, $I n Δ O D A a n d Δ O C A C A = A D (t w o \tan g e n t s f r o m s a m e s i d e) O A = O A (C o m m o n s i d e) O C = O D (R a d i u s o f c i r c l e)$ $\Rightarrow Δ C A O \sim Δ D A O S o, ∠ C O A = ∠ D O A (1)$ Similarly, $I n Δ D O B a n d Δ B O E A ∠ D O B = ∠ E O B (2)$ Since COE is a diameter of the circle, it is a straight line Therefore, $∠ C O A + ∠ D O A + ∠ D O B + ∠ E O B = 180^{0}$ From equations (1) and (2), it can be observed that $2 ∠ D O A + 2 ∠ D O B = 180^{0} \Rightarrow ∠ D O A + ∠ D O B = 180^{0} \Rightarrow ∠ A O B = 90^{0}$ Hence the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre (2)

Given that: TP and TQ are two tangents drawn to a circle with centre O from an external point T To Prove: âˆ PTQ=2âˆ OPQ Construction: join PQ, OP and OQ Proof: âˆ OPT=âˆ OQT=90 In Quadrilateral OPTQ, $∠ P T Q + ∠ O P T + ∠ P O Q + ∠ O Q T = 360^{0} \Rightarrow ∠ P T Q + 90^{0} + ∠ P O Q + 90^{0} = 360^{0} \Rightarrow ∠ P T Q + ∠ P O Q + 180^{0} = 360^{0} \Rightarrow ∠ P T Q + ∠ P O Q = 180^{0} (1)$ $I n Δ O P Q, O P = O Q (r a d i u s o f t h e c i r c l e) ∠ O Q P = ∠ O P Q (e q u a l s i d e s a h v e e q u a l a n g l e s o p p o s i t e t o t h e m) a m d, ∠ P O Q + ∠ O Q P + ∠ O P Q = 180^{0} ∠ P O Q + 2 ∠ O Q P = 180^{0} (2)$ From equation (1) and (2), we have, $∠ P T Q + ∠ P O Q = ∠ P O Q + 2 ∠ O Q P \Rightarrow ∠ P T Q = 2 ∠ O Q P$ Hence Proved

1.prove that the intercept of a tangent between a pair of parallel tangents toa circle substended a right angle at the center of the circle . 2.two tangentsTP andTQ are drawn to a circle with center O, from an external point T.prove that anglePTQ=angleOPQ