Question

in △lmn, ∠n is a right - angle, lm = 76 units, and mn = 40 units. what is the approximate measure of ∠m? a. 40°. b. 32°. c. 28°. d. 58°.

Explanation:

Step1: Recall tangent - ratio formula

In right - triangle $\triangle LMN$ with right - angle at $\angle N$, $\tan\angle L=\frac{MN}{LN}$. But we can also use the inverse - tangent function to find an angle given the ratio of the opposite and adjacent sides. The formula for the inverse - tangent is $\theta=\tan^{- 1}(\frac{\text{opposite}}{\text{adjacent}})$. Here, for $\angle L$, the opposite side to $\angle L$ is $MN = 40$ and the adjacent side is $LN$. However, we can use the fact that $\tan\angle L=\frac{MN}{LN}$ and we want to find $\angle L$. So, $\angle L=\tan^{-1}(\frac{MN}{LM})$.

Step2: Calculate the tangent value

We know that $MN = 40$ and $LM = 76$. Then $\tan\angle L=\frac{MN}{LM}=\frac{40}{76}\approx0.5263$.

Step3: Find the angle using inverse - tangent

Using a calculator, $\angle L=\tan^{-1}(0.5263)$. $\angle L\approx27.7^{\circ}\approx28^{\circ}$.

Answer:

C. $28^{\circ}$