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Mathematics

If (a + b + c + d)(a - b - c - d) = (a + b - c - d)(a - b + c - d);

prove that : a : b = c : d.

Ratio Proportion

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Answer

Given,

(a + b + c + d)(a - b - c - d) = (a + b - c - d)(a - b + c - d)

⇒a+b+c+da+b−c−d=a−b+c−da−b−c+d\Rightarrow \dfrac{a + b + c + d}{a + b - c - d} = \dfrac{a - b + c - d}{a - b - c + d}

Applying componendo and dividendo:

⇒a+b+c+d+a+b−c−da+b+c+d−(a+b−c−d)=a−b+c−d+a−b−c+da−b+c−d−(a−b−c+d)⇒2(a+b)2(c+d)=2(a−b)2(c−d)⇒(a+b)(c+d)=(a−b)(c−d)\Rightarrow \dfrac{a + b + c + d + a + b - c - d}{a + b + c + d - (a + b - c - d)} = \dfrac{a - b + c - d + a - b - c + d}{a - b + c - d - (a - b - c + d)} \\[1em] \Rightarrow \dfrac{2(a + b)}{2(c + d)} = \dfrac{2(a - b)}{2(c - d)} \\[1em] \Rightarrow \dfrac{(a + b)}{(c + d)} = \dfrac{(a - b)}{(c - d)}

Applying alternendo:

⇒(a+b)(a−b)=(c+d)(c−d)\Rightarrow \dfrac{(a + b)}{(a - b)} = \dfrac{(c + d)}{(c - d)}

Applying componendo and dividendo:

⇒a+b+(a−b)a+b−(a−b)=c+d+c−dc+d−(c−d)⇒2a2b=2c2d⇒ab=cd\Rightarrow \dfrac{a + b + (a - b)}{a + b - (a - b)} = \dfrac{c + d + c - d}{c + d - (c - d)} \\[1em] \Rightarrow \dfrac{2a}{2b} = \dfrac{2c}{2d} \\[1em] \Rightarrow \dfrac{a}{b} = \dfrac{c}{d} \\[1em]

Hence, proved that a : b = c : d.

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