find $m\\angle i$.\nwrite your answer as an integer or as a decimal rounded to the nearest tenth.\n$m\\angle…

Explanation:

Step1: Identify sides relative to $\angle I$

In right $\triangle GHI$, hypotenuse $GI=8$, opposite side to $\angle I$ is $GH$. First calculate $GH$ via Pythagorean theorem: $$GH = \sqrt{GI^2 - HI^2} = \sqrt{8^2 - 6^2} = \sqrt{64-36} = \sqrt{28} = 2\sqrt{7}$$

Step2: Use sine function for $\angle I$

$\sin(\angle I) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{GH}{GI}$ $$\sin(\angle I) = \frac{2\sqrt{7}}{8} = \frac{\sqrt{7}}{4}$$

Step3: Calculate $\angle I$ with arcsine

$\angle I = \arcsin\left(\frac{\sqrt{7}}{4}\right)$ Calculate the value: $\frac{\sqrt{7}}{4} \approx \frac{2.6458}{4} \approx 0.6614$, so $\arcsin(0.6614) \approx 41.4^\circ$

Answer:

$41.4$