Question

prove that tangents to a circle at the end points of diameter are equal?​

Answer 1

$\huge\underline\red{\sf CorrectQuestion :-}$

★ Prove that tangents to a circle at the end points of diameter are Parallel ★

Given:

• A circle with center O and diameter AB

• Let PQ be the tangent at point A & RS be the tangent at point B

To Prove: PQ || RS

Since, PQ is a tangent at point A

∴ OA ⟂ PQ [ Tangent at any point of a circle is perpendicular to the radius through point of contact ]

∴ ∠OAP = 90°...........(1)

Similarly,

RS is a tangent at point B

∴ OB ⟂ RS [ Tangent at any point of a circle is perpendicular to the radius through point of contact ]

∴ ∠ OBS = 90°...........(2)

★ From equation (1) and (2) ★

$\small\implies{\sf }$ ∠OAP = 90° and ∠OBS = 90°

Therefore, ∠OAP = ∠OBS i.e ∠BAP = ∠ABS

Now, For lines RS and PQ and transversal AB

$\small\implies{\sf }$ ∠BAP = ∠ABS [ Alternate angles are equal to each other ]

Therefore, The lines PQ is parallel to RS or PQ || RS