To find the length of side \( f \) in triangle \( DEF \) using the Law of Cosines, we can use the formula:
\[ f^2 = d^2 + e^2 - 2de \cdot \cos(F) \]
Given:
- \( d = 10 \)
- \( e = 7 \)
- \( F = 120^\circ \)
First, convert the angle \( F \) from degrees to radians, or simply use the cosine of the angle in degrees:
\[ \cos(120^\circ) = -\frac{1}{2} \]
Now plug the values into the formula:
\[ f^2 = 10^2 + 7^2 - 2 \cdot 10 \cdot 7 \cdot \left(-\frac{1}{2}\right) \]
Calculate each part:
- \( 10^2 = 100 \)
- \( 7^2 = 49 \)
- \( -2 \cdot 10 \cdot 7 \cdot \left(-\frac{1}{2}\right) = 70 \)
Now substitute these back into the equation:
\[ f^2 = 100 + 49 + 70 \]
\[ f^2 = 219 \]
Now take the square root to find \( f \):
\[ f = \sqrt{219} \approx 14.82 \]
Since this value does not exactly match any of the given response options, let's double-check the closest approximate option.
- The closest option to \( 14.82 \) among the provided answers is 14.62 (which is possibly due to rounding).
So based on the choices available, the best answer is:
14.62
1 answer