Question

The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is at least 61 cm, find the minimum length in cm of the shortest side.

Answer 1

Hint: We will start solving this question by assuming the shortest side of a triangle as x cm. then we will get the longest side as 3x and the third side as $3x-2$. Then we will use the formula of perimeter of the triangle and substitute the values. Simplifying the obtained equation we will get the desired answer.

Complete step by step solution:
We have been given that the longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side and the perimeter of the triangle is at least 61 cm.
We have to find the length of the shortest side.
Now, let us assume that the length of the shortest side of a triangle is x cm.
Now, given the question that the longest side of a triangle is 3 times the shortest side so the length of the longest side will be 3x and the third side is 2 cm shorter than the longest side so the length of the third side will be $3x-2$.

Now, we know that the perimeter of the triangle is equal to the sum of all sides of a triangle.
So we will get
$\Rightarrow \text{Perimeter of triangle}=x+3x+\left( 3x-2 \right)$
Now, we have given that the perimeter of the triangle is at least 61 cm, so substituting the value we will get
$\Rightarrow x+3x+\left( 3x-2 \right)\ge 61$
Now, simplifying the above obtained equation we will get
$\begin{align}
  & \Rightarrow x+3x+3x-2\ge 61 \\
 & \Rightarrow 7x\ge 61+2 \\
 & \Rightarrow 7x\ge 63 \\
 & \Rightarrow x\ge \dfrac{63}{7} \\
 & \Rightarrow x\ge 9 \\
\end{align}$
Hence the minimum length of the shortest side must be 9 cm.

Note: Here in this question we used the inequality sign because we have given that the perimeter of the triangle is at least 61 cm. Also it is asked to find the minimum length of the side. The point to be remembered is that do not replace the inequality sign with an equal sign.