for △lmn, lm = 18 in, mn = 12 in, and ln = 15 in. what is the measure of angle l? 82.8° 48.6° 55.7° 41.4°

Explanation:

Step1: Identify the sides and angle

In $\triangle LMN$, we know the lengths of the sides: $LM = 18$ in (adjacent to angle $L$), $LN = 15$ in (adjacent to angle $L$), and $MN = 12$ in (opposite to angle $L$). We can use the Law of Cosines to find angle $L$. The Law of Cosines formula is $c^{2}=a^{2}+b^{2}-2ab\cos(C)$, where $a$, $b$, and $c$ are the sides of the triangle and $C$ is the angle opposite side $c$. Here, side $MN$ (length $12$) is opposite angle $L$, so let $a = LM = 18$, $b = LN = 15$, and $c = MN = 12$. The formula for $\cos(L)$ will be $\cos(L)=\frac{a^{2}+b^{2}-c^{2}}{2ab}$.

Step2: Substitute the values into the formula

Substitute $a = 18$, $b = 15$, and $c = 12$ into the formula for $\cos(L)$: [ \begin{align*} \cos(L)&=\frac{18^{2}+15^{2}-12^{2}}{2\times18\times15}\ &=\frac{324 + 225- 144}{540}\ &=\frac{324+225 = 549; 549 - 144=405}{540}\ &=\frac{405}{540}\ &=\frac{3}{4} = 0.75 \end{align*} ]

Step3: Find the angle whose cosine is 0.75

To find angle $L$, we take the inverse cosine (arccos) of $0.75$: [ L=\arccos(0.75)\approx41.4^{\circ} ]

Answer:

$41.4^{\circ}$ (the option with $41.4^{\circ}$)