Question

. Two circles of unit radius touch each other and each of them touches internally a

circle of radius two, as shown in the figure alongside.

What will be the radius of the smallest circle that

touches all the three circles?​

Answer 1

Answer: 2/3

Step By Step:

We have two circles who touches each other and have unit radius.

Let,

Center of both circles with unit radius be A and B.

So,

Diameter of both circles ED = DF = 2

Now, both circle touches a bigger circle internally that's radius is 2

So,

Diameter of bigger circle EF = 4

And,find the radius of the circle that touches all three circles, As:

Let C is the center of our circle that touches all three (externally to two circles and internally to bigger).

Radius = x

And,

A and B are the center of two circles and D is the center of bigger circle.

In,∆ABC

AB = 1+ 1 = 2

AC = 1+ x

BC = 1+ x

Here AC = BC so ∆ABC is a isosceles triangle,therefore a median from C would bisect AB at D and form 90°.

CD = (Radius of bigger circle) - (Radius of circle that touches all three (x))

CD = 2 - x

Now In ∆ACD

(AC)2 = (AD)2 +(CD)2 (By pythagoras theorem)

After substituting all values we get,

(1+x)2=(1)2+(2−x)2(1+x2+2x) =1 +(4+x2−4x)

6x= 4

⇒ x = 46 = 23

Hope this helps. Please mark me as Brainliest. Have a great day.

Answer 2

The answer is 2/3

Hope it helps