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Under-Specified But Unique |
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Consider a rectangle with two intersecting lines in the interior as shown below. |
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The symbols α, β, γ denote the given areas of the respective regions, and we are asked to determine the remaining area X. This is an example of a problem in which the given information is insufficient to determine the figure, but there is nevertheless a unique answer to the question. To see this, let’s draw vertical and horizontal lines through the central crossing point, and mark the segment lengths as shown below. |
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By similar triangles we have E = (C/D)B and F = (C/D)A, and the given areas lead to the relations |
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We also have the expression for X |
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Lastly, the overall area of the rectangle is |
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The equations for a and γ give |
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Solving the right hand equation for C and substituting from the left hand equation for D, we get |
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Substituting for C and D into the equation for the overall area of the rectangle gives |
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This answers the original question, but it doesn’t determine the values of A,B,C,D. Substituting for C, D, and E in the equation for b, we get the condition |
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Solving for B, this gives |
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Thus for any value of A we get a figure that satisfies the conditions of the problem if we use this value of B along with the values of C and D given by |
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The corresponding values of E and F are |
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This shows that there is a one-parameter family of solutions, one for every value of A, and each of which has the same four areas. Two examples, for A=1 and A=2, are shown below. |
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