Question

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

Answer 1

⇒ Given:

A field in the shape of a quadrilateral.

The measurements are:

PQ = 170 m

QR - 250 m

RS = 100 m

PS = 240 m

PR = 260 m

⇒ To Find:

The area in hectares.

⇒ Solution:

We can divide this quadrilateral into two triangles - ΔPQR and ΔPSR. Then we can find the area of each triangle using Heron's formula. Adding those and dividing that value by 10000, we will get the area in hectares.

In ΔPQR:

PQ = 170 m

QR = 250 m

PR = 260 m

Semiperimeter = $\sf{\dfrac{a+b+c}{2}}$

$\sf{=\dfrac{170+250+260}{2}}$

$\sf{=\dfrac{680}{2}}$

$\sf{=340\:m}$

S - a = 340 - 170 = 170 m

S - b = 340 - 250 = 90 m

S - c = 340 - 260 = 80 m

Area:

$\sf{A=\sqrt{S(S-a)(S-b)(S-c)}}$

$\sf{A=\sqrt{340\times170\times90\times80}}$

$\sf{A=\sqrt{416160000}}$

$\sf{A=20400\:m^2}$

In ΔPRS:

PR = 260 m

RS = 100 m

PS = 240 m

Semiperimeter = $\sf{\dfrac{a+b+c}{2}}$

$\sf{=\dfrac{260+100+240}{2}}$

$\sf{\dfrac{600}{2}}$

$\sf{300\:m}$

S - a = 300 - 260 = 40 m

S - b = 300 - 100 = 200 m

S - c = 300 - 240 = 60 m

$\sf{A=\sqrt{S(S-a)(S-b)(S-c)}}$

$\sf{A=\sqrt{300\times40\times200\times60}}$

$\sf{A=\sqrt{144000000}}$

$\sf{A=12000\:m^2}$

Now, the area of the field:

= Area of ΔPQR + Area of ΔPSR

= 20400 m² + 12000 m²

= 32400 m²

1 hectare = 10000 m

So, 32400 m² is

$\sf{\dfrac{32400}{10000}}$

= 3.24 ha

The required answer is 3.24 ha.

Answer 2

okk

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