Question

The sides of a triangle have lengths $11,15$ and $k,$ where $k$ is an integer. For how many values of $k$ is the triangle obtuse?

Answer 1

Sum of two sides of triangle is more than third side
$\Rightarrow 11+15>k$
$\Rightarrow 26>k\Rightarrow k<26$
difference of any two sides of triangle is less than third side
$\Rightarrow 15-11<k\Rightarrow 4<k\Rightarrow k>4$
$\Rightarrow 4<k<26$
$\Rightarrow k=5,6,7,.....,25$
Given triangle is obtuse,
$\Rightarrow$ either ${11}^2+{15}^2<k^2$ or ${11}^2+k^2<{15}^2$
${11}^2+{15}^2<k^2\Rightarrow k=19,20,21,....,25$
${11}^2+k^2<{15}^2\Rightarrow k=5,6,7,....10$
Number of values possible for $k$ are $7+6=13$