Question

"The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle ."

• A. undefined
• B. undefined
• C. undefined
• D. undefined

Answer 1

Let AB be a chord and angleAOB be the angle subtended by it at centre and angleAPB be the angle subtended at any point P on the remaining part of circle.
Now join PO and extend it to some point Q.
angleAPQ is external angle to triangle POA. $\to$ angleAPQ= angleOPA + angle OAP.
Since OP and OA are radii of circle and hence equal, the triangle OAP is isoceles triangle. Therefore angle OPA = angle OAP.
$\to$ angle APQ = $2\times angleOPA.$ ....(I)
Similarly considering triangle OPB, angle BPQ = $2\times$ angle OPB....(ii)
Adding (i) and (ii),
angle APQ + angle BPQ = $2\times$ (angle OPA + angle OPB)
angle AOB = $2\times$ angle APB ------ PROVED