Question

A surveyor is measuring a triangular plot of land. The lengths of two sides are 150 meters and 200 meters. The angle between these two sides measures 75 degrees. What is the length of the third side of the triangle?

Answer 1

To find the length of the third side of the triangle, we can use the Law of Cosines. The Law of Cosines states:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

where:
- \(a\) and \(b\) are the lengths of the two sides.
- \(C\) is the angle between those sides.
- \(c\) is the length of the side opposite to the angle \(C\).

Here, we have:
- \(a = 150 \, \text{m}\)
- \(b = 200 \, \text{m}\)
- \(C = 75^\circ\)

Now, substituting the values into the formula:

\[
c^2 = 150^2 + 200^2 - 2 \cdot 150 \cdot 200 \cdot \cos(75^\circ)
\]

Calculating \(150^2\) and \(200^2\):

\[
150^2 = 22500
\]
\[
200^2 = 40000
\]

Now adding these values together:

\[
c^2 = 22500 + 40000 - 2 \cdot 150 \cdot 200 \cdot \cos(75^\circ)
\]

Next, we need to compute \(2 \cdot 150 \cdot 200\):

\[
2 \cdot 150 \cdot 200 = 60000
\]

We also need to calculate \(\cos(75^\circ)\). The approximate value of \(\cos(75^\circ)\) is \(0.2588\).

Now substituting this value into the equation:

\[
c^2 = 62500 - 60000 \cdot 0.2588
\]

Calculating \(60000 \cdot 0.2588\):

\[
60000 \cdot 0.2588 \approx 15528
\]

So the equation becomes:

\[
c^2 = 62500 - 15528
\]
\[
c^2 \approx 46972
\]

Finally, taking the square root of both sides to find \(c\):

\[
c \approx \sqrt{46972} \approx 217 \, \text{meters}
\]

Thus, the length of the third side of the triangle is approximately **217 meters**.