Question

In triangle DEF, side d =10, side e = 7, and angle F=120 degrees. What is the length of side f?(1 point)
Responses

15.0
15.0

14.80
14.80

18.0
18.0

14.62

Answer 1

To find the length of side \( f \) in triangle DEF, we can use the Law of Cosines, which states:

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

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

We first need to find \( \cos(120^\circ) \). Since \( 120^\circ \) is in the second quadrant, \( \cos(120^\circ) = -\frac{1}{2} \).

Now we substitute the known 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:

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

\[
= 100 + 49 + 70
\]

\[
= 219
\]

Now, taking the square root:

\[
f = \sqrt{219} \approx 14.83
\]

Among the provided answers, the closest to \( 14.83 \) is \( 14.80 \).

So, the length of side \( f \) is approximately:

\(\boxed{14.80}\)