Question

A triangular field has two sides 60m and 75m Long the angle between them is 102 degree how long is the t

Answer 1

To find the length of the third side of a triangle when given two sides and the angle between them, you can use the Law of Cosines. The Law of Cosines states:

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

Where:
- \(c\) is the length of the side opposite angle \(C\)
- \(a\) and \(b\) are the lengths of the other two sides
- \(C\) is the angle between sides \(a\) and \(b\)

In your case:
- \(a = 60 \, \text{m}\)
- \(b = 75 \, \text{m}\)
- \(C = 102^\circ\)

Now plug in the numbers:

\[
c^2 = 60^2 + 75^2 - 2 \cdot 60 \cdot 75 \cdot \cos(102^\circ)
\]

Calculating each term:

1. \(60^2 = 3600\)
2. \(75^2 = 5625\)
3. Calculate \(\cos(102^\circ)\):

\[
\cos(102^\circ) \approx -0.2079
\]

4. Substitute back into the equation:

\[
c^2 = 3600 + 5625 - 2 \cdot 60 \cdot 75 \cdot (-0.2079)
\]

Calculate \(2 \cdot 60 \cdot 75 = 9000\), then multiply by \(-0.2079\):

\[
9000 \cdot (-0.2079) = -1869.1
\]

Now substituting this into the equation gives:

\[
c^2 = 3600 + 5625 + 1869.1
\]

\[
c^2 = 11094.1
\]

Now, take the square root to find \(c\):

\[
c = \sqrt{11094.1} \approx 105.3 \, \text{m}
\]

So, the length of the third side is approximately **105.3 meters**.