Q16 of 52 Page 10

In fig XP and XQ are tangents from X to the circle with centre O. R is a point on the circle. Prove that, XA+AR+AB+BR. (CBSE 2014)



From the figure, there is an external point X from where two tangents, XP and XQ, are drawn to the circle.
XP = XQ (The lengths of the tangents drawn from an external point to the circle are equal.)
Similarly,
AP = AR
BQ = BR
XP = XA + AP --------- (1)
XQ = XB + BQ --------- (2)
By substituting AP = AR in equation (1) and BQ = BR in equation (2), we get
XP = XA + AR
XQ = XB + BR
Since the tangents XP and XQ are equal, we get
XA + AR = XB + BR.


Hence Proved


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