Question

A triangular plot of land has angles 46° and 58°........

Answer 1

Given the word problem, we can deduce the following information:

1. A triangular plot of land has angles 46° and 58°.

2. The side opposite the 46° is 35 m long.

To determine the length of the fence to enclose the entire plot of land, we make a figure of the triangular plot first:

where:

x= the third angle

a= side opposite of 58°

b= side opposite of x°

Since the sum of the interior angles of a triangle is 180°, we can get the value of x by:

x=180-46-58

x=76°

Next, we use the Law of Sines proportion as shown:

$\frac{\sin46}{35}=\frac{\sin58}{a}=\frac{\sin x}{b}$

Then, we find the value of a:

$\begin{gathered} \frac{\sin46}{35}=\frac{\sin58}{a} \\ \text{Simplify and rearrange} \\ a=\frac{\sin58}{\frac{\sin46}{35}} \\ a=\text{ 41.26 m} \end{gathered}$

We also need to find the value of b:

$\begin{gathered} \frac{\sin46}{35}=\frac{\sin x}{b} \\ \text{Simplify and rearrange} \\ b=\frac{\sin76}{\frac{\sin46}{35}} \\ b=47.21 \end{gathered}$

Hence, we add the sides of which rounded to the nearest meter to get the total length of the fence:

Fence = 35+41+47 =123 meters

Therefore, the length of the fence to enclose the entire plot of land is:

b. 123 meters