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Given that a² + b² = 16² , b² + c² = 24² , c² + d² = 42² and d² + a² = 38² then find max value of product of (a + c) and (b + d).
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(Shortcut Image Solution with Calculations is gives below in this post)
This Algebraic question can be solved Geometrically,
It's diagram will be:
AB = a
BD = b
BC = c
BE = d
AD = 16
DC = 24
CE = 42
AE = 38
Note (a + c)(b + d) will give area of this figure since AC and DE are perpendicular to each other.
Here we can apply Ptolemy's Theorem to calculate product of diagonal.
(AC)(DE) = (AD)(CE) + (CD)(AE)
⇒ (a + c)(b + d) = (16)(42) + (24)(38) = 1584
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