Question

Show that the angles subtended by an arc at the centre is double the angle subtended by same arc at any point in the remaining part of the circle

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

Answer 1

Let AB be a chord and angle AOB be the angle subtended by it at centre and angle APB be the angle subtended at any point P on the remaining part of circle.
Now join PO and extend it to some point Q.
angle APQ is external angle to triangle POA. $\to$ angle APQ = angle OPA + 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 $\times$ angle OPA...(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