Question

The lengths of the sides of a triangle are in integers and its area is also an integer. One side is $21$ cm and the perimeter is $48$ cm, then the length of the shortest side is

• A. $8$ cm
• B. $10$ cm
• C. $12$ cm
• D. $14$ cm

Answer 1

Answer: (B)

$10$ cm

To solve these type of question, take the help of the options given.
You know,
$a+b+c=48$
$a+b=27$

$s=\frac{a+b+c}{2}=\frac{48}{2}=24$
Now take each option as a value of $a$ and then find the area using,
$A=\sqrt{s(s-a)(s-b)(s-c)}$

Let $a=8⟹b=19$

$⟹A=\sqrt{24(24-8)(24-19)(24-21)}$

$⟹A=\sqrt{5760}⟹A$ is not an integer.

Let $a=10⟹b=14$

$A=\sqrt{24(24-10)(24-17)(24-21)}$

$⟹A=\sqrt{7056}=84$ is an integer.
Thus, length of shortest side is $10$ cm.

Hence, option B is correct.