Angles can be classified into five groups, based on their measure in degrees.
- Acute: angles with measure $< 90^\circ$
- Right: angles with measure $= 90^\circ$
- Obtuse: angles with measure $> 90^\circ$ and $< 180 ^ \circ$
- Straight: angles with measure $= 180^ \circ$
- Reflex: angles with measure $> 180^ \circ$> and $< 360 ^ \circ$
If $a, b,$ and $c$ are the $3$ sides of an acute triangle where $c$ is the longest side then $c^{2} < a^{2} + b^{2}$
The sides are $10, 12,$ and $'x'.$
Case$1:$ Among the $3$ sides $10, 12,$ and $x,$ for values of $x \leq 12,\; 12 $ is the longest side.
When $x \leq 12,$ let us find the number of values for $x$ that will satisfy the inequality $12^{2} < 10^{2} + x^{2}$
$\implies144 < 100 + x^{2}$
$\implies 44<x^{2}$
$\implies x^{2}>44$
The least integer value of $x$ that satisfies this inequality is $7.$
The inequality will hold true for $x = 7, 8, 9, 10, 11,$ and $12$. i.e., $6$ values.
Case$2:$ For values of $x > 12, x$ is the longest side.
Let us count the number of values of $x$ that will satisfy the inequality $x^{2} < 10^{2} + 12^{2}$
i.e., $x^{2} < 244$
$x = 13, 14,$ and $15$ satisfy the inequality. That is $3$ more values.
Hence, the values of $x$ for which $10, 12,$ and $x$ will form sides of an acute triangle are $x = 7, 8, 9, 10, 11, 12, 13, 14, 15.$
So, the correct answer is $(C).$
