Q. 64.5( 21 Votes )

In figure, AB and CD are common tangents to two circles of equal radii. Prove that AB = CD.


Answer :

Given: AB and CD are two common tangents to two circles of equal radii .


To Prove: AB = CD



Construction: Join OA, OC, O’B and O’D


Proof:


Now, OAB = 90° and OCD = 90° as OA AB and OC CD


[tangent at any point of a circle is perpendicular to radius through the point of contact]


Thus, AC is a straight line.


Also,


O'BA = O'DC = 90° [Tangent at a point on the circle is perpendicular to the radius through point of contact]


Thus, BD is Also a straight line.


So ABCD is a quadrilateral with Four sides as AB, BC, CD and AD


But as


A = B = C = D = 90°


So, ABCD is a rectangle.


Hence, AB = CD [opposite sides of rectangle are equal]


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