Question

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
OR
A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC.

Answer 1

Given : A circle (O,r) and a tangent I at point A.
To prove : $OA\perp L$
Construction : take any point B other than A on the tangent. Join OB.
Suppose OB meets the cirlce at C.
Proof : Amoung all line segements joining the centre O to any point on l, the perpendicular is the shortest to l.
So, in order to prove $OA\perp$ we need to prove that OA is shorter than OB.
OA =OC (Radius of same circle)
Now, OB =OC +BC