Question

The shape of a farm is a quadrilateral. Measurements taken of the farm, by naming its corners as P, Q, R, S in order are as follows. l(PQ) = 170 m, l(QR) = 250 m, l(RS) = 100 m, l(PS) = 240 m, l(PR) = 260 m.
Find the area of the field in hectare (1 hectare = 10,000 sq.m)

Answer 1

In $∆$PSR,
$$\begin{gathered} s=\frac{100+260+240}{2}=\frac{600}{2}=300 \\ \mathrm{Ar}\left(△\mathrm{PSR}\right)=\sqrt{300\left(300-100\right)\left(300-260\right)\left(300-240\right)} \\ =\sqrt{300\times 200\times 40\times 600} \\ =\sqrt{144000000} \\ =12000\mathrm{sq}\mathrm{m} \end{gathered}$$
In $∆$PQR,
$$\begin{gathered} s=\frac{260+250+170}{2}=340 \\ \mathrm{Ar}\left(△\mathrm{PQR}\right)=\sqrt{340\left(340-260\right)\left(340-250\right)\left(340-170\right)} \\ =\sqrt{340\times 80\times 90\times 170} \\ =\sqrt{416160000} \\ =20400\mathrm{sq}\mathrm{m} \end{gathered}$$
Area of PQRS = Ar($∆$PSR) + Ar($∆$PQR)
= 12000 + 20400
= 32400 m²
1 hectare = 10,000 sq.m
$32400{\mathrm{m}}^2=3.24\mathrm{hec}$