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Let L represent the length of the rectangle and W represent the width.
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\n" ); document.write( "Since the perimeter (P) of a rectangle is the distance around it, the perimeter can be
\n" ); document.write( "written in equation form as:
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\n" ); document.write( "P = L + W + L + W
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\n" ); document.write( "and when like terms are combined, this simplifies to:
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\n" ); document.write( "P = 2L + 2W
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\n" ); document.write( "You are told that the perimeter is 28, so you can substitute 28 for P to make the equation:
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\n" ); document.write( "28 = 2L + 2W
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\n" ); document.write( "Next notice that when you draw a diagonal of the rectangle, the diagonal divides the rectangle
\n" ); document.write( "into 2 right triangles. Each of these triangles has as one of its legs L and as the other
\n" ); document.write( "leg W, and it has as its hypotenuse (longest side) the diagonal (D). Since both triangles
\n" ); document.write( "are right triangles, the Pythagorean theorem applies ... that is the sum of the squares of
\n" ); document.write( "the legs of one of these triangles equals the square of the hypotenuse. And since the
\n" ); document.write( "legs of one of the triangle are L and W and the hypotenuse is D, we can write the Pythagorean
\n" ); document.write( "equation as:
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\n" ); document.write( "But the problem tells us that the diagonal, D, equals 10. Therefore
\n" ); document.write( "and we can substitute that value into the Pythagorean equation in place of
\n" ); document.write( "get:
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\n" ); document.write( "Now let's return to the perimeter equation and solve it for one of the variables in terms of
\n" ); document.write( "the other. The perimeter equation is:
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\n" ); document.write( "28 = 2L + 2W
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\n" ); document.write( "Divide both sides (all terms) by the common factor of 2 and you reduce this equation to:
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\n" ); document.write( "14 = L + W
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\n" ); document.write( "Let's subtract L from both sides to get rid of it on the right side and we get:
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\n" ); document.write( "14 - L = W
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\n" ); document.write( "Now we can substitute 14 - L for W in the Pythagorean equation because 14 - L and W are equals.
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\n" ); document.write( "Begin with the Pythagorean equation:
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\n" ); document.write( "and replace W with 14 - L to get:
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\n" ); document.write( "Square out the expression in the parentheses and this equation becomes:
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\n" ); document.write( "On the left side, combine the squared terms and arrange the terms in descending powers of L
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\n" ); document.write( "Just to make things a little easier, note that 2 is a common factor of all terms, so
\n" ); document.write( "we can divide this entire equation (all terms) by 2 to get:
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\n" ); document.write( "To get this into the more conventional form, get rid of the 50 on the right side by subtracting
\n" ); document.write( "50 from both sides to reduce the equation to:
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\n" ); document.write( "Notice that the left side of this equation can be factored into:
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\n" ); document.write( "and that this equation will be true if either of the two factors on the left side equals
\n" ); document.write( "zero ... because multiplication by zero on the left side will make the left side equal to
\n" ); document.write( "the zero on the right side.
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\n" ); document.write( "So either
\n" ); document.write( "that either L = 8 or L = 6.
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\n" ); document.write( "If L = 8, then we can go to the perimeter equation of
\n" ); document.write( "8 for L to get:
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\n" ); document.write( "14 = 8 + W
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\n" ); document.write( "Subtracting 8 from both sides results in W = 6.
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\n" ); document.write( "But if L = 6 and we go to the same perimeter equation, when we substitute 6 for L we get:
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\n" ); document.write( "14 = 6 + W
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\n" ); document.write( "and subtracting 6 from both sides we get W = 8. But this can't be because by definition
\n" ); document.write( "the width (W) is shorter than the length and in this case the width is 8 and the length is 6.
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\n" ); document.write( "So the valid solution is the one in which we found the length is 8 and the width is 6.
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\n" ); document.write( "Just as a check we can add 8 + 6 + 8 + 6 and find that the perimeter is 28, as the problem
\n" ); document.write( "said it should be. And we can apply the Pythagorean theorem:
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\n" ); document.write( "and squaring the terms on the left side results in:
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\n" ); document.write( "This simplifies to:
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\n" ); document.write( "and taking the square root of both sides:
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\n" ); document.write( "So when L = 8 and W = 6, the diagonal is 10, just as the problem said it is.
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\n" ); document.write( "Our answer checks out.
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\n" ); document.write( "Hope this helps you to understand the problem and one method by which the answer can be found.
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