In the figure if AC is a common tangent of two touching circles. If B is touching point of both circles...

Question

In the figure if AC is a common tangent of two touching circles. If B is the touching point of both circles then which is of the following is true about it and why...

1. Angle B is a right angle
2. Angle B is an obtuse angle
3. Angle B is an acute angle
4. There is not enough information to know anything about Angle B.

Answer 1

Suppose O1 is center of small circle and O2 of bigger one. Now O1 A is perpendicular at A on AC and O2C is perpendicular at C on the tangent AC. If I join O1 O2 then angle O1AC = angle O1AB + angle BAC = 90. Similarly, angle O2CA = angle O2CB + angle BCA.
This means angle B = angle A + angle C so B is 90.

Answer B is right angled.