Question

In triangle DEF, side d =10, side e = 7, and angle F=120 degrees. What is the length of side f?

Answer 1

To find the length of side \( f \) in triangle \( DEF \) where \( d = 10 \), \( e = 7 \), and angle \( F = 120^\circ \), we can use the Law of Cosines, which states:

\[
f^2 = d^2 + e^2 - 2de \cos(F)
\]

Plugging in the given values:

- \( d = 10 \)
- \( e = 7 \)
- \( F = 120^\circ \)

First, calculate \( \cos(120^\circ) \):

\[
\cos(120^\circ) = -\frac{1}{2}
\]

Now substitute the values into the Law of Cosines formula:

\[
f^2 = 10^2 + 7^2 - 2 \cdot 10 \cdot 7 \cdot \left(-\frac{1}{2}\right)
\]

Calculating each term:

\[
f^2 = 100 + 49 + 2 \cdot 10 \cdot 7 \cdot \frac{1}{2}
\]

Calculating the last term:

\[
2 \cdot 10 \cdot 7 \cdot \frac{1}{2} = 10 \cdot 7 = 70
\]

Now, plug this back into the equation:

\[
f^2 = 100 + 49 + 70
\]

\[
f^2 = 219
\]

Taking the square root of both sides gives:

\[
f = \sqrt{219}
\]

Thus, the length of side \( f \) is:

\[
f \approx 14.8
\]

So, the length of side \( f \) is \( \sqrt{219} \) or approximately \( 14.83 \).