Question

Find the area of a parallelogram whose two adjacent sides are 50cm and 40cm respectively and diagonal is 30 cm

Answer 1

We are given a parallelogram in which we are given that the adjacent sides are 40cm and 50cm respectively.

One of its diagonals be 30cm.

As shown in the figure attached, let ABCD be the parallelogram.

And as we know that corresponding sides are parallel and equal in a parallelogram .

$\Rightarrow AB = CD = 50\ cm\\\Rightarrow BC = AD = 40\ cm\\\text{Also, diagonal }AC = 30\ cm$

Also, the each diagonals divide the parallelogram in 2 equal area triangles.

And, if we find the area of one such triangle and multiply it by 2, we can find the area of parallelogram.

$\Rightarrow \text{Area of parallelogram ABCD} = 2 \times \text{ area of } \triangle ABC$

We can use heron's formula to find area of $\triangle ABC$:

$\text{Area of a }\triangle = \sqrt{s(s-a)(s-b)(s-c)}$

where a,b and c are the sides of $\triangle$;

s is called the semi-perimeter of $\triangle$

Formula for semi-perimeter is:

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

As per the attached figure, using heron's formula  on $\triangle ABC$

$s = \dfrac{30+40+50}{2} \Rightarrow s = 60\\\text {Area of }\triangle ABC = \sqrt{60(60-30)(60-40)(60-50)}\\\Rightarrow \sqrt{60(30)(20)(10)}\Rightarrow 600 cm^{2}$

Hence, Area of parallelogram = $2 \times 600 = 1200\ cm^{2}$