Question

hey guys..❤


Q :

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)


Thanks in advance..

Answer 1

$\textbf{Solution}$

ANSWER : $\textbf{3.24 hectares}$

$\textbf{STEP-BY-STEP EXPLANATION}$

For ∆ PQR,

$l \: (pq) = a = 170m \\ \\ l \: (qr) = b = 250m \\ \\ l \: (pr) = c = 260m \\ \\ semiperimeter \: (s) \: = \: \frac{a + b + c}{2} \\ \\ = \: \frac{170 + 250 + 260}{2} \\ \\ = \frac{680}{2} \\ \\ = 340m.$

A ( ∆PQR )

$= \: \sqrt{s(s - a) \: (s - b) \: (s - c)} \\ \\ = \: \sqrt{340(340 - 170) \: (340 - 250) \: (340 - 260)} \\ \\ = \: \sqrt{340 \times 170 \times 90 \times 80} \\ \\ = \: \sqrt{170 \times 2 \times 170 \times 9 \times 10 \times 80} \\ \\ = \: \sqrt{170 \times 170 \times 9 \times 2 \times 10 \times 8 \times 10} \\ \\ = \: \sqrt{170 \times 170 \times 9 \times 16 \times 10 \times 10} \\ \\ = \: \sqrt{ {170}^{2} \times {3}^{2} \times {4}^{2} \times {10}^{2} } \\ \\ = \: 170 \times 3 \times 4 \times 10 \\ \\ = \: \textbf{20400 \: sq \: m}$

For ( ∆PSR )

$l(ps) = {a}^{1} = \: 240m \\ \\ l(sr) = {b}^{1} = \: 100m \\ \\ l(pr) = {c}^{1} = \: 260m \\ \\ semiperimeter \: ( {s}^{1})= \: \frac{ {a}^{1} \: + {b}^{1} \: + {c}^{1} }{2} \\ \\ = \: \frac{240 + 100 + 260}{2} \\ \\ = \: \textbf{300m }$

A ( ∆PSR )

$= \sqrt{ {s}^{1} \: ( {s}^{1} - {a}^{1} )\: (s - {b}^{1} ) \: ( {s}^{1} - {c}^{1} ) } \\ \\ = \sqrt{300 \: (300 - 240) \: (300 - 100) \: (300 - 260)} \\ \\ = \sqrt{300 \times 60 \times 200 \times 40} \\ \\ = \: \sqrt{3 \times 100 \times 3 \times 20 \times 20 \times 10 \times 40} \\ \\ = \: \sqrt{3 \times 3 \times 100 \times 20 \times 20 \times 400} \\ \\ = \: \sqrt{ {3}^{2} \times {10}^{2} \times {20}^{2} \times {20}^{2} } \\ \\ = \: 3 \times 10 \times 20 \times 20 \\ \\ = \: \textbf{12000 \: sq \: m }$

A ( ∆PQRS )

= A (∆PQR) + A (∆PSR)

= 20400 + 12000

= 32400 sq m

= $\textbf{3.24 hectares}$

( › 1 hectare = 10000 sq m )

$\textbf{Ans : Area of the field is 3.24 hectares}$