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# The longest side of a triangle is three times the shortest side and third side is $2\;cm$ shortest than the longest side. If the perimeter of the triangle is at least $61\; cm$ find the minimum length of the shortest side.

$\begin{array}{1 1} (A)\;9\;cm \\(B)\;3\;cm \\(C)\;5\;cm \\(D)\;5\;cm \end{array}$

Toolbox:
• Same Quantity can be added (a subtracted ) to (from ) both sides of the inequality with out changing the sign of the in equality.
• Same positive quantities can be multiplied or divided to both side of the in equality with out changing the sign of the inequality.
• If same negative quantity is multiplied or divided to both sides of the inequality is reversed i.e $'>'$ sign changes to $'<'$ and $'<'$ changes $'>'$ .
Let the length of the shortest side be $x\;cm$
Length of the largest side is $3x\;cm$
Length of the third side is $3x-2\;cm$
Since the perimeter of the triangle is at least 61 cm, we get,
$x +3x +3x-2 \geq 61$
$=> 7x -2 \geq 61$
$=> 7x \geq 61+2$
$7x \geq 63$
Dividing both sides by positive number 7.
$\large\frac{7x}{7} \geq \frac{63}{7}$
$x \geq 9$
Step 2:
The minimum length of the shortest side is 9 cm.
Hence A is the correct answer.