Question

A field in the form of a parallelogram has sides 60 m and 40 m and one of its diagonals is 80 m long.
The area of the parallelogram is $600\sqrt{k}{m}^2$. Find the value of $k$

Answer 1

From above figure,

Quadrilateral $ABCD$ is a parallelogram, with sides $AB=CD=60m,BC=DA=40m$ and diagonal $BD=80m$

Area of parallelogram $ABCD=2\times$ (Area of $△$ABD)

In $△$ABD,

Semi-perimeter of $△ABD$,

$s=\frac{a+b+c}{2}$

$\Rightarrow$$\frac{AB+BD+DA}{2}$

$\Rightarrow$$\frac{60+80+40}{2}$

$\Rightarrow$$\frac{180}{2}$

$\Rightarrow 90m$

So, by Heron's formula,

Area of $△$ABD = $\sqrt{s(s-a)(s-b)(s-c)}$

$\Rightarrow$$\sqrt{90(90-60)(90-80)(90-40)}$

$\Rightarrow$ $\sqrt{90\times 30\times 10\times 50}$

$\Rightarrow$$100\times 3\sqrt{15}{m}^2$

So, area of parallelogram $ABCD=2\times 300\sqrt{15}=600\sqrt{15}{m}^2$

$\therefore$ The value of $k$ is $15$.