Question

A piece of land is in the shape of a rhombus ABCD in which each side measures 100 m and diagonal AC is 160 m long. Find the area of the rhombus.

Answer 1

$$\begin{gathered} \mathrm{Let}: \\ a=100\mathrm{m},b=160\mathrm{m}\mathrm{and}c=100\mathrm{m} \\ s=\frac{a+b+c}{2}=\frac{100+160+100}{2}=180\mathrm{m} \\ \mathrm{By}\mathrm{Heron}\text{'}\mathrm{s}\mathrm{formula},\mathrm{we}\mathrm{have}: \\ \mathrm{Area}\mathrm{of}\mathrm{triangle}ABC=\sqrt{s(s-a)(s-b)(s-c)} \\ =\sqrt{180(180-100)(180-160)(180-100)} \\ =\sqrt{180\times 80\times 20\times 80} \\ =\sqrt{20\times 3\times 3\times 80\times 20\times 80} \\ =20\times 3\times 80 \\ =4800{\mathrm{m}}^2 \end{gathered}$$

We know that a diagonal divides a parallelogram into two triangles of equal areas.
∴ Area of parallelogram ABCD = 2(Area of triangle ABC) = $2\times 4800=9600{\mathrm{m}}^2$