triangle A B C has different side lengths a, b, c. If a2, b2, c2 as sides form another triangle P Q R, then triangle A B C will always be

Question

triangle A B C has different side lengths a, b, c. If a2, b2, c2 as sides form another triangle P Q R, then triangle A B C will always be (A) acute angled triangle only (B) obtuse angled triangle only (C) sometimes acute or sometimes obtuse depending on values of a, b and c (D) none of these

Tardigrade Admin · Accepted Answer

Let a > b > c, given that b+c > a and b2+c2 > a2 ∴ b2+c2-a2 > 0 ∴ 2 b c cos A > 0 ∴ cos A > 0 ∴ A is acute angle. Similarly cos B > 0, cos C > 0 So, triangle A B C is an acute angled triangle.

Q. $△ A BC$ has different side lengths $a, b, c$ . If $a^{2}, b^{2}, c^{2}$ as sides form another $△ PQR$ , then $△ A BC$ will always be

acute angled triangle only

obtuse angled triangle only

sometimes acute or sometimes obtuse depending on values of $a, b$ and $c$

none of these

Let $a > b > c$ , given that $b + c > a$ and $b^{2} + c^{2} > a^{2}$
$∴ b^{2} + c^{2} - a^{2} > 0$
$∴ 2 b c cos A > 0$
$∴ cos A > 0$
$∴ A$ is acute angle.
Similarly $cos B > 0, cos C > 0$
So, $△ A BC$ is an acute angled triangle.

Q. △ABC has different side lengths a,b,c. If a2,b2,c2 as sides form another △PQR, then △ABC will always be