All changes saved 14. In \( \triangle L M N, L M=12, L N=10 \), and \( m \angle L=52^{\circ} \). What is the length, to the nearest tenth of a unit, of \( \overline{M N} \) ? 7.4 units 14.4 units 9.8 units 15.6 units (i)

To find the length of \( \overline{M N} \) in triangle \( \triangle L M N \), we can use the Law of Cosines, which states that \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \). Here, we identify \( L M = a = 12 \), \( L N = b = 10 \), and \( \angle L = C = 52^{\circ} \). Plugging the values into the formula gives us: \[ MN^2 = 12^2 + 10^2 - 2 \times 12 \times 10 \times \cos(52^\circ) \] Calculating that will yield a number we can then take the square root of to find \( MN \). After calculating, you'll arrive at approximately \( 9.8 \) units for \( \overline{M N} \). This triangle scenario is a fantastic way to explore trigonometry in real life! The Law of Cosines is especially handy in navigation, where understanding distances between points is crucial. For instance, sailors or pilots can rely on this theorem to chart their course accurately when dealing with angled paths, ensuring they reach their destinations efficiently. If you're diving deeper into triangle relationships, don't overlook the world of triangle inequalities! These rules help determine possible triangle conditions and can prevent those "oops" moments where you think a triangle can exist, but it really can't. Remember, the sum of the lengths of any two sides must always be greater than the length of the third side. Happy triangle adventuring!