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.

Answer 1

(i) Perimeter of the triangular field = 324 m

If the unknown side is x m in length, we have:

$$\begin{gathered} 85+154+x=324 \\ \Rightarrow x=324-154-85=85 \end{gathered}$$

Also,
Semiperimeter of the triangular field, $s=\frac{\mathrm{Perimeter}}{2}=\frac{324}{2}=162\mathrm{m}$

Now, the area of the field can be expressed as:

$$\begin{gathered} A=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)} \\ =\sqrt{162\left(162-154\right)\left(162-85\right)\left(162-85\right)} \\ =\sqrt{162\times 8\times 77\times 77} \\ =\sqrt{7683984} \\ =2772{\mathrm{m}}^2 \end{gathered}$$


(ii) Area of the triangle can be expressed as:

$$\begin{gathered} A=\frac{1}{2}\times b\times h \\ \Rightarrow 2772=\frac{1}{2}\times 154\times h \\ \Rightarrow h=\frac{2772\times 2}{154}=36\mathrm{m} \end{gathered}$$

Thus, the length of the perpendicular from the opposite vertex on the side measuring 154 m is 36 m.