Question

The sides of a $△ABC$ are positive integers. The smallest side has length $1$ . Which of the following statements is true?

• A. The area of $△ABC$ is always a rational number
• B. The area of $△ABC$ is always an irrational number
• C. The perimeter of $△ABC$ is an even integer
• D. The information provided is not sufficient to conclude any of the statements $A,B$ or $C$ above

Answer 1

Answer: (B)

We have, sides of $△ABC$ are positive integer and length of smallest side is $1.$

We know that the sum of two sides of triangle is greater than third side.
$\therefore b+1>c$
$\Rightarrow c-b<1$
$1+c>b$
$b-c<1$
$-1<b-c<1$
$b,c$ are integers.
$\therefore b-c=0\Rightarrow b=c$
Semi-perimeter $=\frac{a+b+c}{2}$
$=\frac{2b+1}{2}=b+\frac{1}{2}$
Area$A=\sqrt{s(s-a)(s-b)(s-c)}$
$=\sqrt{\left(b+\frac{1}{2}\right)\left(b+\frac{1}{2}-1\right)\left(b+\frac{1}{2}-b\right)\left(b+\frac{1}{2}-b\right)}$
$=\sqrt{\left(b+\frac{1}{2}\right)\left(b-\frac{1}{2}\right)\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)}=\frac{1}{2}\sqrt{b^2-\frac{1}{4}}$
Since, $b$ is integer.
$\therefore$ Area of $△ABC$ is always irrational number.