Question

If $x$ and $y$ are two non-collinear vectors and $A B C$ is a triangle with side lengths $a, b, c$ satisfying $(20 a-15 b) x+(15 b-12 c) y+(12 c-20 a)(x \times y)=\vec{0}$ then $\triangle A B C$ is

• A. an acute-angled triangle
• B. an obtuse-angled triangle
• C. a right-angled triangle
• D. an isosceles triangle

Answer 1

Answer: (C)

Since $x$ and $y$ are linearly independent,
$20 a-15 b$
$=15 b-12 c$
$=12 c-20 a=0 $
$\Rightarrow \frac{a}{3}=\frac{b}{4}=\frac{c}{5}$
$\Rightarrow c^{2}=a^{2}+b^{2}$
$ \Rightarrow \Delta A B C$ is right-angled.