Question

The distance between the centers of the three circles, which are mutually tangent with each other, are 10,12 and 14 units, Find the area of the largest circle.

Answer 1

@jim_thompson5910 can you help me ? :)

Answer 2

let me think

Answer 3

This is a tough problem and I had to do a bit of searching. I found this though

https://www.ics.uci.edu/~eppstein/junkyard/tangencies/three-circles.html

which helps construct the three circles

This is what I get when I follow the steps in that article. See attached. Again I used geogbera to help me out

Answer 4

As for how to do this without using a visual aid like this, I'm not sure. I have a feeling it will rely on using coordinate geometry somehow

Answer 5

it can be solved with a simple system of equations

Answer 6

@Zarkon do you mean this?

x+y = 14
y+z = 10
x+z = 12

where x,y,z are the radii of the circles. I've thought about it but I overlooked it because it seemed too easy, but idk

Answer 7

yes

Answer 8

ok then go with that method @SnitchSeeker1496

Answer 9

x+y = 14
y+z = 10

solve the second equation for y
y = 10-z
then use substitution

x+10-z = 14
x-z = 4

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So we now have these two equations
x-z = 4
x+z = 12

use elimination to solve

Answer 10

x =16, y= -2, z = 12 ?

Answer 11

@jim_thompson5910 after getting the the values of x, y and z, what should I do next?

Answer 12

`x =16, y= -2, z = 12 ?` is incorrect unfortunately

Answer 13

Oh wait, x is for the largest circle right? so the answer is 64 pi for the area?

Answer 14

it sounds like you got 8 to be the largest radius which is correct, so yes, the largest circle has area 64pi

Answer 15

i forgot to divide it by 2 sorry :))

Answer 16

btw you should find that...
x = 8
y = 6
z = 4

Answer 17

x = 8, y = 6 , z =4 right?

Answer 18

correct

Answer 19

Thank you :)

Answer 20

no problem