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If $10$, $12$ and '$x$' are sides of an acute angled triangle, how many integer values of '$x$' are possible ?

1. $7$
2. $12$
3. $9$
4. $13$

Option C. 9

Acute Angle : between 0 and 89 degrees.

Case 1: X is the longest side

then X < $\sqrt{10^{2}+12^{2}}$ i.e. X < 16

Case 2: 12 is the longest side.

then X >  $\sqrt{12^{2}-10^{2}}$  i.e.  X > 6

Therefore x ranges from 7 to 15 = 9 Values

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Suppose, if here it was given that there are two sides known to us,i.e 10 and 12, then how many acute-angled triangles are possible to form ?? How do you deal with this question now

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 $(C).$

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