Question

Find the areas of the given plot. (All measures are in metres.)

Answer 1

In ∆ABE, m∠BAE = 90°, l(AB) = 24 m, l(BE) = 30 m

∴ [l(BE)]² = [l(AB)]² + [l(AE)]² …[Pythagoras theorem]

∴ (30)² = (24)² + [l(AE)]²

∴ 900 = 576 + [l(AE)]²

∴ [l(AE)]² = 900 – 576

∴ [l(AE)]² = 324

∴ l(AE) = ${\displaystyle \sqrt{324}}$

= 18 m …[Taking square root of both sides]

A(∆ABE) = ${\displaystyle \left(\frac{1}{2}\right)}$ x product of sides forming the right angle

= ${\displaystyle \left(\frac{1}{2}\right)\times l\left(\text{AE}\right)\times l\left(\text{AB}\right)}$

= ${\displaystyle \left(\frac{1}{2}\right)\times 18\times 24}$

= 216 sq. m

In ∆BCE, a = 30m, b = 28m, c = 26m

Semiperimeter of ∆BCE,

S =  ${\displaystyle \frac{1}{2}}$ (a + b + c) 

= ${\displaystyle \frac{30+28+26}{2}}$

= ${\displaystyle \frac{84}{2}}$

= 42 m

A (∆BCE) = ${\displaystyle \sqrt{s(s-a)(s-b)(s-c)}}$

= ${\displaystyle \sqrt{42(42-30)(42-28)(42-26)}}$

= ${\displaystyle \sqrt{42\times 12\times 14\times 16}}$

= ${\displaystyle \sqrt{6\times 7\times 6\times 2\times 2\times 7\times 8\times 2}}$

= ${\displaystyle \sqrt{6^2\times 7^2\times 8^2}}$

= ${\displaystyle 6\times 7\times 8}$

= 336 sq. m

In ∆EDC, l(CE) = 28 m, l(DF) = 16 m

A(∆EDC) = ${\displaystyle \frac{1}{2}\times \text{base}\times \text{height}}$

${\displaystyle \frac{1}{2}\times l\left(\text{CE}\right)\times l\left(\text{DF}\right)}$

= ${\displaystyle \left(\frac{1}{2}\right)\times 28\times 16}$

= 224 sq. m.

∴ Area of plot ABCDE

= A(∆ABE) + A(∆BCE) + A(∆EDC)

= 216 + 336 + 224

= 776 sq. m

∴ The area of the given plot is 776 sq.m.