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| TriNRect (Posted on 2010-10-13) |
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Let R be the area of rectangle ABCD and
T the area of triangle APQ (where P and
Q are points on sides BC and CD of ABCD
respectively).
What is the minimal value of |BP|+|DQ|
in terms of R and T?
Note: R and T are constants with 0 < T ≤ R/2.
Therefore, |BP| and |DQ| are not independent variables
(i.e. if P varies between B and C, then Q
must vary between C and D such that T
stays constant).
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Submitted by Bractals
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Solution:
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Let w = |AB|, x = |BP|, y = |DQ|, and z = |AD|.
Let [FG...KL] denote the area of polygon FG...KL.
Then
[ABCD] = [APQ] + [ABP] + [ADQ] + [CPQ]
or
R = T + ½wx + ½yz + ½(w-y)(z-x)
or
xy = R - 2T
To minimize the value of |BP|+|DQ| = x + y,
we let x = y and get
2√(R-2T).
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