prove that the tangents to a circle at the end points of a diameter are parallel
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To prove: PQ∣∣ RS
Given: A circle with centre O and diameter AB. Let PQ be the tangent at point A & Rs be the point B.
Proof: Since PQ is a tangent at point A.
OA⊥ PQ(Tangent at any point of circle is perpendicular to the radius through point of contact).
∠OQP=90o …………(1)
OB⊥ RS
∠OBS=90o ……………(2)
From (1) & (2)
∠OAP=∠OBS
i.e., ∠BAP=∠ABS
for lines PQ & RS and transversal AB
∠BAP=∠ABS i.e., both alternate angles are equal.
So, lines are parallel.
$$\therefore PQ||RS.
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