14) State and prove Tangent perpendicularity theorem
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A tangent to a circle is perpendicular to the radius through the point of contact.
Given :- A circle C (O, r) and a tangent AB at a point P.
To prove :- OP is perpendicular to AB
Construction :- Take any point Q, other than P, on the tangent AB. Join OQ. Suppose OQ meets the circle at R.
Proof :- We know that among all line segments joining the point O to a point on AB, the shortest one is perpendicular to AB. So, to prove that OP is perpendicular to AB, it is sufficient to prove that OP is shorter than any other segment joining O to any point on AB.
Now, OP = OR (radii of the same circle)
Also, OQ = OR +RQ
⇒ OQ > OR
⇒ OQ >OP (∵ OP =OR)
⇒ OP < OQ
Thus, OP is shorter than any segment joining OP to any point on AB.
Hence, OP is perpendicular to AB.
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