find the tangent of $angle i$.\nsimplify your answer and write it as a proper fraction, improper fraction…

Explanation:

Step1: Find side FG via Pythagoras

In right $\triangle FGH$, $FG = \sqrt{FH^2 - GH^2} = \sqrt{45^2 - 27^2}$ $\sqrt{2025 - 729} = \sqrt{1296} = 36$

Step2: Confirm similar triangles

$\triangle FGH \sim \triangle EGI$ (AA similarity, right angle + shared $\angle I$). So $\frac{EG}{GI} = \frac{FG}{GH}$

Step3: Define GI as $27+x$, EG as $36+y$

But $\tan(I) = \frac{FG}{GH} = \frac{EG}{GI}$, so use $\triangle FGH$ values: $\tan(I) = \frac{\text{opposite}}{\text{adjacent}} = \frac{FG}{GH}$

Step4: Simplify the fraction

$\tan(I) = \frac{36}{27} = \frac{4}{3}$

Answer:

$\frac{4}{3}$