Ttangents PQ and PR are drawn at a circle with centre O from an external point P prove that - Maths - Circles

Question

Ttangents PQ and PR are drawn at a circle with centre O from an
external point P prove that <QPR = 2 <OQR

Lalit Mehra · Accepted Answer

Hi!
Here is the answer to your question
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Given: A circle with centre O PQ and PR are tangents drawn
from an external point P to the circle
To prove: ∠QPR = 2∠OQR
Proof:
∠OQP = ∠ORP = 90° (Radius is perpendicular to the
tangent at point of contact)
In quadrilateral OQPR
∠QPR + ∠OQP + ∠QOR + ∠ORP = 360°
∴∠OPR + 90° + ∠QOR + 90° =
360°
⇒ ∠OPR + ∠QOR = 180° (1)
In âˆ†OQR
OQ = OR (Radius of the circle)
⇒ ∠ORQ = ∠OQR (Equal sides have equal angles
opposite to them)
∠QOR + ∠OQR + ∠ORQ = 180° (angle
sum property)
⇒ ∠QOR + 2∠OQR = 180° (2)
From (1) and (2), we get
∠QPR + ∠QOR = ∠QOR + 2∠OQR
⇒ ∠QPR = 2∠OQR
Cheers!

Ttangents PQ and PR are drawn at a circle with centre O from an external point P prove that <QPR = 2 <OQR