In fig. 7, two equal circles, with centers O and O’, touch each other at X. OO’ produced meets the circle with center O’ at A. AC is tangent to the circle with center O, at the point C. O’D is perpendicular to AC. Find the value of  (CBSE 2016)

Given : Two equal circles with center O and O' , touch each other at X. AC is tangent to the circle center O, at the point C and O'D⏊OC

To Find :

OD⏊AC

∠ODC = 90°

∠OCD = 90° [The tangent drawn at a point on a circle is perpendicular to the radius through the point of radius]

∠ODC + ∠OCD = 180°

O'D || OC

[ If a transversal intersect two lines and interior angles on the same side of transversal are supplementary, lines are parallel ]

Let the radius of both circles is r.

Then, AO' = r

AO = AO' + O'X + XO

= r + r + r = 3r

Now,

[If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.]