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प्रश्न
In the figure, XP and XQ are tangents from X to the circle with centre O. R is a point on the circle. Prove that XA + AR = XB + BR.
योग
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उत्तर
Given:
XP and XQ are tangents from an external point X to a circle with centre O.
RRR is a point on the circle.
The line XR meets the tangents at points A and B respectively.
We have to prove:
XA + AR = XB + BR
Tangents drawn from an external point to a circle are equal in length.
XP = XQ ...(1)
Segment addition on straight lines
XP = XA + AP
XQ = XB + BQ
Substitute from (1)
XA + AP = XB + BQ ...(2)
Equal tangent segments from a point on the circle
Since R lies on the circle:
AP and AR are tangents from point A
BQ and BR are tangents from point B
AP = AR and BQ = BR
XA + AR = XB + BR
Hence Proved.
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