37
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
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Correct figure [$\frac{1}{2}$ mark]
Let AB be the diameter of the circle.
Since tangent $\perp$ radius at point of contact
therefore $\angle OAP = \angle OBR = 90^\circ$ [$\frac{1}{2}$ mark]
$\Rightarrow \angle OAP + \angle OBQ = 180^\circ$ [$\frac{1}{2}$ mark]
$\Rightarrow$ Co-interior angles are supplementary
$\Rightarrow PQ \parallel RS$
Hence tangents at the ends of a diameter of a circle are parallel. [$\frac{1}{2}$ mark]
Let AB be the diameter of the circle.
Since tangent $\perp$ radius at point of contact
therefore $\angle OAP = \angle OBR = 90^\circ$ [$\frac{1}{2}$ mark]
$\Rightarrow \angle OAP + \angle OBQ = 180^\circ$ [$\frac{1}{2}$ mark]
$\Rightarrow$ Co-interior angles are supplementary
$\Rightarrow PQ \parallel RS$
Hence tangents at the ends of a diameter of a circle are parallel. [$\frac{1}{2}$ mark]