Question

In a triangle, if the sum of two sides is $x$ and their product is $y$ such that $(x+z)(x-z)=y$ where $z$ is the third side of triangle then the triangle is

• A. equilateral
• B. right angled
• C. obtuse angled
• D. isosceles

Answer 1

The correct option is C obtuse angled

Let ABC be the triangle with $b+c=x$ and $bc=y$ then $a=z$,

given $\Rightarrow (x+z)(x-z)=y$

$\Rightarrow (b+c+a)(b+c-a)=bc$

$\Rightarrow (b+c{)}^2-a^2=bc$

$\Rightarrow b^2+c^2-a^2=-bc$

$\Rightarrow \frac{b^2+c^2-a^2}{2bc}=-\frac{1}{2}$

$\Rightarrow cosA=-\frac{1}{2}=cos{120}^o$

$\Rightarrow A={120}^o$

$\therefore$ the triangle is obtuse angled.