Question

Prove that if an arc of a circle subtends a right angle at any point on the remaining part of the circle, then the arc is a semi-circle.

Answer 1

Given : A circle with centre O and an arc AB subtending $\angle$ACB at a point C on the remaining part of the circle such that $\angle$ACB = ${90}^{\circ }$.

To Prove: ArcAB is a semi-circle.
Construction : Join OA and OB

STATEMENT REASON
1. Arc AB subtends $\angle$AOB at the centre
and $\angle$ACB at a point C on the
remaining part of the circle.

$\angle$AOB = $\angle$ACB .............. (i) Angle at the centre is double the angle at a point on the remaining part of the circle.

2. $\angle$ACB = ${90}^{\circ }$ ..........(ii) Given

3. $\angle$AOB = (2 x ${90}^{\circ }$) = ${180}^{\circ }$ From (i) and (ii)
AOB is a straight line

AOB is a diameter Chord AB passes through the centre O.

Arc AB is a semi-circle.