Question

The sides of a triangle have length $11,15$ and $k$, where $k$ is an integer. Then, the number of values of $k$ for which the triangle is obtuse is

• A. $5$
• B. $12$
• C. $13$
• D. $17$

Answer 1

Answer: (C)

$13$

Using the rules, sum of two sides should always be greater than third side and modulus difference of two sides will less than the third side, we get $4<k<26$

By cosine rule, ${11}^2+{15}^2<k^2$ or ${11}^2+k^2<{15}^2$
$\Rightarrow 336<k^2$ or $k^2<104$
Also,
$k<11+15\Rightarrow k<26$ and $11+k>15\Rightarrow k>4$
$\Rightarrow k\in$ {$5,6,7,8,9,10,19,20,21,22,23,24,25$}