Two sides of a triangular field are 85 m and 154 m in length and its perimeter is 324 m

Question:

Two sides of a triangular field are 85 m and 154 m in length and its perimeter is 324 m. Find (i) the area of the field and (ii) the length of the perpendicular from the opposite vertex on the side measuring 154 m.

Solution:

(i) Let:

$a=85 \mathrm{~m}$ and $b=154 \mathrm{~m}$

Given :

Perimeter $=324 \mathrm{~m}$

or, $a+b+c=324$

$\Rightarrow c=324-85-154=85 \mathrm{~m}$

$\therefore s=\frac{324}{2}=162 \mathrm{~m}$

By Heron's formula, we have:

Area of triangle $=\sqrt{s(s-a)(s-b)(s-c)}$

$=\sqrt{162(162-85)(162-154)(162-85)}$

$=\sqrt{162 \times 77 \times 8 \times 77}$

$=\sqrt{1296 \times 77 \times 77}$

$=\sqrt{36 \times 77 \times 77 \times 36}$

$=36 \times 77$

$=2772 \mathrm{~m}^{2}$

(ii) We can find out the height of the triangle corresponding to 154 m in the following manner:
We have:

Area of triangle $=2772 \mathrm{~m}^{2}$

$\Rightarrow \frac{1}{2} \times$ Base $\times$ Height $=2772$

$\Rightarrow$ Height $=\frac{2772 \times 2}{154}=36 \mathrm{~m}$

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