If a, b, c are the sides of a non-equilateral triangle, then the expre

1.	If a, b, c are the sides of a non-equilateral triangle, then the expression (b + c – a) (c + a – b) (a + b – c) – abc is(a) negative (b) non-negative (c) positive (d) non-positive
Answer» (a) negative Let x = b + c – a, y = c + a – b, z = a + b – c Since a, b, c are the sides of a non-equilateral triangle, by the triangle inequality a + b > c, b + c > a, c + a > b ⇒ a + b – c > 0, b + c – a > 0, c + a – b > 0 ⇒ z > 0, x > 0, y > 0 Also, x, y, z are distinct. ∴ AM > GM \(c=\frac{x+y}{2}>\sqrt{xy}\) \(b=\frac{x+z}{2}>\sqrt{xz}\) \(a=\frac{y+z}{2}>\sqrt{yz}\) ∴ abc > \(\sqrt{xy}.\sqrt{xz}.\sqrt{yz}\) ⇒ abc > xyz ⇒ abc > (b + c – a) (c + a – b) (a + b – c) ⇒ (b + c – a) (c + a – b) (a + b – c) – abc < 0, i.e., negative.

If a, b, c are the sides of a non-equilateral triangle, then the expression (b + c – a) (c + a – b) (a + b – c) – abc is(a) negative (b) non-negative (c) positive (d) non-positive

Answer»

(a) negative

Let x = b + c – a, y = c + a – b, z = a + b – c

Since a, b, c are the sides of a non-equilateral triangle, by the triangle inequality a + b > c, b + c > a, c + a > b

⇒ a + b – c > 0, b + c – a > 0, c + a – b > 0

⇒ z > 0, x > 0, y > 0

Also, x, y, z are distinct.

∴ AM > GM

\(c=\frac{x+y}{2}>\sqrt{xy}\)

\(b=\frac{x+z}{2}>\sqrt{xz}\)

\(a=\frac{y+z}{2}>\sqrt{yz}\)

∴ abc > \(\sqrt{xy}.\sqrt{xz}.\sqrt{yz}\)

⇒ abc > xyz ⇒ abc > (b + c – a) (c + a – b) (a + b – c)

⇒ (b + c – a) (c + a – b) (a + b – c) – abc < 0, i.e., negative.