Israel 2012 Q1 - Tangent circles in rectangle

Art of Problem Solving

AoPS Online

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

No tags match your search

M

Israel 2012 Q1 - Tangent circles in rectangle

G H J

Reveal topic tags

tangent circles

L

G H BBookmark kLocked kLocked NReply

Source: Israel National Olympiad 2012 Q1

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

71 posts

Cuubic

#1 h Aug 8, 2019, 8:50 PM • 2 Y

Y by Adventure10, Mango247

In the picture below, the circles are tangent to each other and to the edges of the rectangle. The larger circle's radius equals 1. Determine the area of the rectangle.

https://i.imgur.com/g3GUg4Z.png

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

2351 posts

#2 h Aug 8, 2019, 9:17 PM • 2 Y

Y by Adventure10, Mango247

The 2 smaller circles have radii $\frac 12$ since the width of the rectangle equals the diameter of the big circle ( $2$ ) which equals the combined diameters of the 2 smaller circles. Now we need to find the length of the rectangle. By connecting the centers of the 3 circles, we form an isosceles triangle with base length $1$ and equal side lengths of $1+\frac 12=\frac 32$ . The length of the rectangle is equivalent to the sum of the height of the isosceles triangle plus the radius of the smaller circles and the radius of the largest circle. We can draw the altitude from the top of the isosceles triangle to the base. By Pythagoras, we have: $\bigg(\frac 12\bigg)^2+x^2=\bigg(\frac 32\bigg)^2\implies x^2=2\implies x=\sqrt 2.$ The length of the rectangle is simply: $\sqrt 2+\frac 12+1=\frac{3+2\sqrt 2}{2}.$ We finalize the area of the rectangle: $2\bigg(\frac{3+2\sqrt 2}{2}\bigg)=\boxed{3+2\sqrt 2}~~\blacksquare$
Is this incorrect? I don't know why but feel like my solution is incomplete/incorrect.

This post has been edited 2 times. Last edited by kittenwarrior, Aug 8, 2019, 11:29 PM

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

71 posts

Cuubic

#3 h Aug 8, 2019, 10:34 PM • 1 Y

Y by Adventure10

I believe you are correct, except for the final result in the box where you forgot to multiply by 2

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

2351 posts

#4 h Aug 8, 2019, 11:28 PM • 2 Y

Y by Adventure10, Mango247

Oops, sorry lol, idk how to multiply xD

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a