Question

Two adjacent sides of a parallelogram are $5cm$ and $3.5cm$. One of its diagonals is $6.5cm$. Then the area of parallelogram is

• A. $5\sqrt{3}$ $c{m}^2$
• B. $10\sqrt{3}$ $c{m}^2$
• C. $15\sqrt{3}$ $c{m}^2$
• D. $20\sqrt{3}$ $c{m}^2$

Answer 1

Answer: (B)

$10\sqrt{3}$ $c{m}^2$

Given, $ABCD$ is a parallelogram having $AB=5cm,BC=3.5cm$ and one of the diagonal $AC=6.5cm$.
We know that the diagonal of a parallelogram divides it into two triangles of equal area.
Therefore, in triangle $ABC$, by heron's formula, we have $a=5cm,b=3.5cm$ and $c=6.5cm$
$\Rightarrow$ semi-perimeter of triangle $ABC,s=\frac{a+b+c}{2}=\frac{5+3.5+6.5}{2}=7.5cm$

$\therefore$ Area of triangle $ABC=\sqrt{s(s-a)(s-b)(s-c)}=\sqrt{7.5(7.5-5)(7.5-3.5)(7.5-6.5)}=5\sqrt{3}{cm}^2$

$\Rightarrow$ Area of parallelogram $ABCD=2\times \text{area of triangle}ABC=2\times 5\sqrt{3}{cm}^2=10\sqrt{3}{cm}^2$