Question

find gh. write your answer as an integer or as a decimal rounded to the nearest tenth. gh = submit

Explanation:

Step1: Identify the triangle type and trigonometric ratio

We have a right - triangle \( \triangle FGH \) with \( \angle F = 90^{\circ} \), \( \angle G=44^{\circ} \), and the length of the side opposite to \( \angle G \) (i.e., \( FH=\sqrt{30}\)) and we want to find the hypotenuse \( GH \).

In a right - triangle, the sine of an angle \( \theta \) is defined as \( \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} \). Here, \( \theta = \angle G = 44^{\circ} \), the opposite side to \( \angle G \) is \( FH=\sqrt{30}\), and the hypotenuse is \( GH \). So, \( \sin(44^{\circ})=\frac{FH}{GH} \), which can be rewritten as \( GH=\frac{FH}{\sin(44^{\circ})} \).

Step2: Calculate the value of \( FH \)

First, calculate the numerical value of \( FH=\sqrt{30}\approx5.477 \).

Step3: Calculate the value of \( GH \)

We know that \( \sin(44^{\circ})\approx0.6947 \).

Substitute \( FH\approx5.477 \) and \( \sin(44^{\circ})\approx0.6947 \) into the formula \( GH = \frac{FH}{\sin(44^{\circ})} \):

\( GH=\frac{\sqrt{30}}{\sin(44^{\circ})}\approx\frac{5.477}{0.6947}\approx7.9 \)

Answer:

\( 7.9 \)