Question

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

Explanation:

Step1: Identify triangle type and trigonometric ratio

We have a right - triangle \( \triangle GIH \) with \( \angle I = 90^{\circ} \), \( \angle G=41^{\circ} \), and \( IH = 6 \). We want to find the length of the hypotenuse \( GH \). We can use the sine function, where \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \). For \( \angle G \), the opposite side to \( \angle G \) is \( IH \) and the hypotenuse is \( GH \). So \( \sin(41^{\circ})=\frac{IH}{GH} \).

Step2: Solve for \( GH \)

We know that \( IH = 6 \) and \( \sin(41^{\circ})\approx0.6561 \). From \( \sin(41^{\circ})=\frac{6}{GH} \), we can re - arrange the formula to solve for \( GH \): \( GH=\frac{6}{\sin(41^{\circ})} \). Substitute \( \sin(41^{\circ})\approx0.6561 \) into the formula: \( GH=\frac{6}{0.6561}\approx9.1 \)

Answer:

\( 9.1 \)