Question

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

Answer 1

To find the length of the third side of the triangle when two sides and the included angle are known, 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 known sides,
- \( C \) is the angle between those two sides,
- \( c \) is the length of the side opposite angle \( C \).

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

Now, we can plug in the values:

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

Calculating \( 60^2 \) and \( 75^2 \):

\[
60^2 = 3600, \quad 75^2 = 5625
\]

Now sum these:

\[
c^2 = 3600 + 5625 - 2 \cdot 60 \cdot 75 \cdot \cos(102^\circ)
\]

Next, we need to calculate \( \cos(102^\circ) \). Using a calculator:

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

Now we substitute this value back into the equation:

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

Calculating \( 2 \cdot 60 \cdot 75 \approx 9000 \):

\[
c^2 = 3600 + 5625 + 9000 \cdot 0.2079
\]

Calculating \( 9000 \cdot 0.2079 \):

\[
9000 \cdot 0.2079 \approx 1869.1
\]

Now substituting this back in:

\[
c^2 = 3600 + 5625 + 1869.1 = 11094.1
\]

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

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

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