Question

If the equations below hold true for triangles ABC and DEF: \[ a(a + b + c) = d^2, b(a + b + c) = e^2, c(a + b + c) = f^2 \] Then which of the following is always true of triangle DEF?

• A. It is an acute-angled triangle.
• B. It is a right-angled triangle.
• C. It is an obtuse-angled triangle.
• D. None of the above.

Hint:

Whenever you see equations linking side squares to sum expressions, try expressing in ratios or normalizing to find familiar identities like Pythagoras.

Answer 1

The Correct Option is B

We are given: \[ a(a + b + c) = d^2, \quad b(a + b + c) = e^2, \quad c(a + b + c) = f^2 \] Let \( S = a + b + c \). Then: \[ d^2 = aS, \quad e^2 = bS, \quad f^2 = cS \Rightarrow \frac{d^2}{a} = \frac{e^2}{b} = \frac{f^2}{c} = S \] Now, consider: \[ d^2 + e^2 + f^2 = aS + bS + cS = S(a + b + c) = S^2 \Rightarrow d^2 + e^2 + f^2 = S^2 \] From earlier: \[ \frac{d^2}{a} = \frac{e^2}{b} = \frac{f^2}{c} = S \Rightarrow \text{Ratios are preserved} \] So, the side lengths \( d, e, f \) of triangle DEF satisfy: \[ d^2 + e^2 = f^2 \quad \text{(for example)} \Rightarrow \text{Triangle is right-angled} \] Final Answer: \( \boxed{\text{Right-angled triangle}} \)