Find the Perimeter of the Rectangle

Date: 09/06/2002 at 17:08:11
From: Booram
Subject: Circles and rectangles

This question is too hard for me. Can you please tell me the solution?

Two circles of radii 9 and 17 centimetres are enclosed within a 
rectangle with one side of length 50 cm. The two circles touch each 
other, and each touches two adjacent sides of the rectangle. Find the 
perimeter.

Please help me!

Date: 09/06/2002 at 18:05:53
From: Doctor Greenie
Subject: Re: Circles and rectangles

Hello, Booram -

Cool problem!

The problem is actually ambiguous the way you state it; in order to 
get a "nice" solution, we need to assume that one circle touches one 
pair of adjacent sides of the rectangle and the other circle touches 
the other pair of adjacent sides (in other words, we don't have the 
case where both circles touch the same side of the rectangle).

I will describe my figure using a Cartesian coordinate system.

Let the side of the rectangle OABC with length 50 be OA, with O(0,0) 
and A(50,0). Then the opposite side of the rectangle is BC with B 
(50,y) and C(0,y). We will have the problem solved if we can 
determine the value of y, thus fixing the other dimension of the 
rectangle.

Let P(9,9) be the center of the circle with radius 9, so that it 
touches sides OA and OC of the rectangle. Let Q be the center of the 
circle with radius 17; because it is going to touch sides AB and BC 
of the rectangle, it is 17 units to the left of AB and 17 units below 
BC, so its coordinates are Q(33,y-17).

Now if you draw triangle PQR with R(33,9), triangle PQR is a right 
triangle with right angle at R. The unknown dimension of the rectangle 
is 9 + QR + 17; so we will be done with the problem if we 
can determine the length of QR.

But this we can do, because PQR is a right triangle, and we know the 
lengths of PQ and PR.

See if you can see the path to the solution using the method sketched 
above.  Write back if you have any further questions on any of this.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/