Question

Equal circles with centres O and O' touch each other at X. OO' produced to meet a circle with centre O', at A. AC is a tangent to the circle whose centre is O. O'D is perpendicular to AC. Find the value of $\frac{DO\text{'}}{CO}$.

Answer 1

Consider the two triangles and .

We have,

is a common angle for both the triangles.

(Given in the problem)

(Since OC is the radius and AC is the tangent to that circle at C and we know that the radius is always perpendicular to the tangent at the point of contact)

Therefore,

From AA similarity postulate we can say that,

~

Since the triangles are similar, all sides of one triangle will be in same proportion to the corresponding sides of the other triangle.

Consider AO′ of and AO of .

Since AO′ and O′X are the radii of the same circle, we have,

AO′ = O′X

Also, since the two circles are equal, the radii of the two circles will be equal. Therefore,

AO′ = XO

Therefore we have

Since ~,

We have found that,

Therefore,