Question

The lengths of the sides of $∆ABC$ are consecutive integers. If $∆ABC$ has the same perimeter as an equilateral triangle with a side of length $9cm$, what is the length of the shortest side of $∆ABC$?

• A. $6$
• B. $4$
• C. $8$
• D. $10$

Answer 1

The correct option is C

$8$

Explanation for the correct option:

Find the perimeter of the triangle $∆ABC$

Given the side of the equilateral triangle is $9cm$.

The perimeter of the equilateral triangle will be $9cm+9cm+9cm=27cm$.

Since it is given that the perimeter of $∆ABC$ is equal to the perimeter of the equilateral triangle.

So, the perimeter of $∆ABC$ is $27cm$.

Find the length of the shortest side

Since it is given that the sides of the triangle are consecutive integers.

Now, let the sides are $xcm,x+1cm,x+2cm$.

Therefore, the perimeter of $∆ABC$$=x+x+1+x+2=27cm$

$$\begin{gathered} x+x+1+x+2=27 \\ \Rightarrow 3x+3=27 \\ \Rightarrow 3x=24 \\ \Rightarrow x=8cm \end{gathered}$$

Thus, the shortest side $=8cm$.

Hence, the correct option is (c) $8cm$.