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/
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