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 \( DEF \) given side \( d = 10 \), side \( e = 7 \), and angle \( f = 120^\circ \), we can use the Law of Cosines. The Law of Cosines states:

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

In this case, we need to substitute \( d = 10 \), \( e = 7 \), and \( F = 120^\circ \).

First, we calculate \( \cos(120^\circ) \):
\[
\cos(120^\circ) = -\frac{1}{2}
\]

Now we can substitute 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:
\[
10^2 = 100
\]
\[
7^2 = 49
\]
\[
2 \cdot 10 \cdot 7 \cdot \left(-\frac{1}{2}\right) = -70
\]

Putting it all together:
\[
f^2 = 100 + 49 + 70
\]
\[
f^2 = 219
\]

Now, take the square root to find \( f \):
\[
f = \sqrt{219}
\]

Calculating the approximate value:
\[
f \approx 14.8
\]

Thus, the length of side \( f \) is approximately \( 14.8 \) units.