Hello, dumluck!

$\displaystyle \text{If }10, 12\text{ and }x\text{ are sides of an acute triangle,}$

. . $\displaystyle \text{how many integer values of }x\text{ are possible?}$

. . $\displaystyle (a)\;7 \qquad (b)\;12 \qquad (c)\;9 \qquad (d)\;13 \qquad (e)\;11$

$\displaystyle \text{Rule 1: For an acute triangle, the square of the longest side must}$

. . . . . . . . . $\displaystyle \text{be }less\:than\text{ the sum of squares of the other two sides.}$

$\displaystyle \text{Rule 2: For any triangle, sum of any two sides must be}$

. . . . . . . . . $\displaystyle \text{greater than the third side.}$

Code:

*
* *
* * 12
10 * *
* *
* *
* * * * * * *
x

From Rule 1: .$\displaystyle x^2 \:<\:10^2 + 12^2 \quad\Rightarrow\quad x^2 \:<\<244 \quad\Rightarrow\quad x \:<\:16 $

From Rule 2: .$\displaystyle 10 + x\:>\:12 \quad\Rightarrow\quad x \:>\:2$

Hence: .$\displaystyle x \,\in\, \{3,4,5,6,7,8,9,10,11,12,13,14,15\}$

$\displaystyle \,x$ can have 13 integer values . . . answer (d).