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Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Properties of Triangles

Olympiad workbook · IOQM · Properties of Triangles

a, b, c are positive real numbers such that a^2+b^2 =c^2 and a b=c...

a, b, c

are positive real numbers such that

a^2+b^2 =c^2

and

a b=c

. Determine the value of \left|\frac{(a+b+c)(a+b-c)(b+c-a)(c+a-b)}{c^2}\right|

.

Correct answer

04

Step-by-step solution

\begin{aligned} & a^2+b^2=c^2, a b=c \\ & (a+b)^2=a^2+b^2+2 a b=c^2+2 c \\ & (a+b)^2-c^2=2 c ...(i)\\ & (a-b)^2=c^2-2 c \Rightarrow c^2-(a-b)^2=2 c ...(ii) \end{aligned}

Given expression is

\frac{\left((a+b)^2-c^2\right)\left(c^2-(a-b)^2\right)}{c^2}=\frac{2 c \times 2 c}{c^2}=4

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