Course / Category - 9th

Concept Detail
Maths / Circles / Angle in a Semicircle
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Theorem 1  The angle in a semi- circle is a right angle.

 Given     PQ is a diameter of a circle C(O,r) and $\dpi{120} \large \angle PRQ$  is an angle in semi- circle.To prove  $\dpi{120} \large \angle PRQ=90^{\circ}$PROOF  We know that the angle subtended by an arc of a circle at its centre is twice the angle formed by the same arc at a point on the circle. So, we have              $\dpi{120} \large \angle POQ=2\angle PRQ$$\dpi{120} \large \Rightarrow \;\;\;180^{\circ}=2\angle PRQ$                [$\dpi{120} \large \because POQ$  is a straight line]$\dpi{120} \large \Rightarrow \;\;\;\angle PRQ=90^{\circ}$

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