Question

The lengths of the sides of a triangle are in integers and its area is 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

The correct option is B 10 cm

Let the sides of triangle are x,27-x and 21

And x<27-x _x<14

it is a well known fact that the sum of any side of triangle is greater then the other side

So $x+21>27+x\to x>3$

Therefore the possible value of x are 4 to 13

Now apply heron formula

$S=\frac{(a+b+c)}{2}=\frac{48}{2}=24$

A=\sqrt{24(24-x)(x-3)(3))}= \sqrt{72(x-3)(24-x)}=6\sqrt{2(x-3)(24-x)}$

The possible value of x are from 4 to 13

Of these only x=10 will give an integer of area

So A is 10 cm