CBSE Class 9 Answered
Prove that the angle subtended by an arc at
the center is double the angle subtended by it at any point on the remaining
part of the circle.
Asked by Topperlearning User | 04 Jun, 2014, 01:23: PM
Expert Answer
Given:
A circle with centre O in which PQ is any arc which makes 
POQ at the centre and PRQ on the circle at R.
To
Prove: POQ = 2PRQ
Construction: Join RO and produce which meet the circle at point S.
Proof
: In 
OPR
OP = OR (Radii of the same circle)
1 = 3 (Opposite angles of equal sides)
Exterior
5 = 1 + 3 (by exterior angle theorem)
5
= 1 + 1
5
= 21
...(1)
Similarly,
6 = 22
...(2)
On adding equation (1) and (2)
5
+ 6 = 21 + 22
POQ
= 2[1 + 2]
POQ
= 2PRQ
Answered by | 04 Jun, 2014, 03:23: PM
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