Find radius of circle

Geometry Level 3

Two circles of unit radii touch each other and each of them touches internally a circle of radius two,as shown in figure. The radius of the circle which touches all three circles is __________ \text{\_\_\_\_\_\_\_\_\_\_} .


The answer is 0.66.

Ankit Alria by ANKIT ALRIA , 20, India

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3 solutions

Satyabrata Dash
Feb 29, 2016

6 Helpful 0 Interesting 0 Brilliant 0 Confused

I used the same method....

Hritesh Mourya - 5 years, 3 months ago

By pythagorean theorem, we have

( r + 1 ) 2 = h 2 + 1 (r+1)^2=h^2+1

r 2 + 2 r + 1 = h 2 + 1 r^2+2r+1=h^2+1

r 2 + 2 r = h 2 r^2+2r=h^2

h = r 2 + 2 r h=\sqrt{r^2+2r}

Hence,

2 = h + r = r 2 + 2 r + r 2=h+r=\sqrt{r^2+2r}+r

2 − r = r 2 + 2 r 2-r=\sqrt{r^2+2r}

Square both sides.

( 2 − r ) 2 = ( r 2 + 2 r ) 2 (2-r)^2=(\sqrt{r^2+2r})^2

4 − 4 r + r 2 = r 2 + 2 r 4-4r+r^2=r^2+2r

4 = 2 r + 4 r 4=2r+4r

r = 2 3 ≈ 0.66667 r=\dfrac{2}{3}\approx 0.66667

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Akash Singh
Feb 17, 2016

drop perpendicular from O to AB as D

OB = 1+r

DB =1

so find OD

and OD + r=2

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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