Question

Prove that the sum of the squares of the lengths of sides of a rectangle is equal to the sum of the squares of the lengths of its diagonals

Answer 1

Here, it is a rectangle. So opposite sides are equal, and diagonals are also equal. Also all angles are 90° each.

Assume a rectangle ABCD with A at bottom left, B at bottom right, C at top right, and D at top left. Please ask me if you've doubt.

Here, AB = CD, so AB² = CD², and BC = AD, so BC² = AD².

∴ AB² + BC² = CD² + AD²     →     (1)

And also, AC = BD,  so   AC² = BD²     →     (2)

Consider ΔABC.

Draw an altitude BM from B to AC at M.

Consider ΔABC and ΔBMC.

∠ABC = ∠BMC = 90°

∠ACB = ∠MCB  (common)

∴ ΔABC ~ ΔBMC

∴ AC / BC = BC / MC

⇒ BC² = AC × MC     →     (3)

Consider ΔABC and ΔAMB.

∠ABC = ∠AMB = 90°

∠CAB = ∠MAB  (common)

∴ ΔABC ~ ΔAMB

∴ AC / AB = AB / AM

⇒ AB² = AC × AM     →     (4)

(4) + (3)

⇒ AB² + BC² = AC × AM  +  AC × MC

⇒ AB² + BC² = AC(AM + MC)

⇒ AB² + BC² = AC × AC

⇒ AB² + BC² = AC²     →     (5)

From (5),

⇒ AB² + BC² = AC²

⇒ 2 (AB² + BC²)  =  2 (AC²)

⇒ AB² + BC² + AB² + BC² = AC² + AC²

⇒ AB² + BC² + CD² + AD² = AC² + BD²     (Substituting (1) and (2))

Hence proved!

Hope this helps you. ^_^

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