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In figure, AB and CD are common tangents...

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

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Given AB and CD are tangents to two circles of equal radii.
To prove AB=CD

Construction Join OA, OC, O' B and O'D
Proof Now, `angleOAB=90^(@)`
[tangent at any point of a circle is perpendicular to radius through the point of contact]
Thus, AC is a straight line.
Also, `angleOAB+angleOAB=180^(@)`
`:.AB//CD`
Similarly, BD is a straight line
and `angleO'BA=angleO'DC=90^(@)`
Also, AC=BD [radii of two circles are equal]
In quadrilateral ABCD, `angleA=angleB=angleC=angleD=90^(@)`
and AC=BD
ABCD is a rectangle
Hence, AB=CD [opposite sides of rectangle are equal]
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