Question

# In the given figure, AB and CD are common tangents to two circles of unequal radii. if radii of the two circles are equal ,prove that AB=CD.

Solution

## Given : Two circles of equal radii, two common tangents, AB and CD on circles, $$C_1$$ and $$C_2$$.To prove : $$AB = CD$$Construction : Join $$O_1 A, O_1 C$$ and $$O_2 B$$ and $$O_2D$$ . Also join $$O_1 O_2$$.Proof : Since tangent at any point of a circle is perpendicular to the radius to the point of contact.$$\therefore \angle O_1 AB = \angle O_2 BA = 90^{\circ}$$As $$O_1A = O_2B$$, so $$O_1 ABO_2$$ is a rectangleSince opposite sides of a rectangle are equal $$\therefore AB = O_1 O_2$$ ___(i)Similarly, we can prove that $$O_1 CDO_2$$ is a rectangle.$$\therefore O_1O_2 = CD$$ ___(ii)From (i) and (ii) , we get$$AB = CD$$Hence proved.Maths

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