Final answer:
To find the length of side LM in triangle ALMN, the angle ∠N is initially calculated using the other two angles. The Law of Sines is then used to determine the length of LM, which, after calculation, gives a value to the nearest tenth of an inch.
Step-by-step explanation:
The student is asking about finding the length of side LM in triangle ALMN using the given angles ∠L and ∠M, and the length of side n (LMN). To find the length of LM, we need to use the Law of Sines which states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. In this case, we do not have all the angles of the triangle as angle ∠N is missing. First, we need to find ∠N using the fact that the sum of angles in a triangle is 180°.
∠N = 180° - ∠L - ∠M = 180° - 64° - 31° = 85°.
Now we can apply the Law of Sines:
\(rac{LM}{sin(N)} = \frac{n}{sin(L)}\)
\(rac{LM}{sin(85°)} = \frac{910}{sin(64°)}\)
LM = \(sin(85°) \cdot \frac{910}{sin(64°)}\)
After calculating, we find the length LM to the nearest tenth.