Question

find gi.
write your answer as an integer or as a decimal rounded to the nearest tenth.
gi = \boxed{}

Explanation:

Step1: Identify triangle type and trigonometric ratio

We have a right - triangle \( \triangle GIH \) with \( \angle I = 90^{\circ} \), \( \angle G=55^{\circ} \), and hypotenuse \( GH = \sqrt{34} \). We want to find the length of \( GI \), which is adjacent to \( \angle G \). The cosine of an angle in a right - triangle is defined as \( \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} \). So, \( \cos(55^{\circ})=\frac{GI}{GH} \).

Step2: Substitute the known values

We know that \( GH = \sqrt{34}\approx5.83095 \) and \( \cos(55^{\circ})\approx0.5736 \). Substituting into the formula \( GI = GH\times\cos(55^{\circ}) \), we get \( GI=\sqrt{34}\times\cos(55^{\circ}) \).

First, calculate \( \sqrt{34}\approx5.831 \). Then, multiply by \( \cos(55^{\circ})\approx0.5736 \): \( 5.831\times0.5736\approx3.345 \approx 3.3 \) (rounded to the nearest tenth).

Alternatively, we can use the cosine formula directly:
\( \cos(55^{\circ})=\frac{GI}{\sqrt{34}} \)
\( GI=\sqrt{34}\cos(55^{\circ}) \)
\( \sqrt{34}\approx5.83095 \)
\( \cos(55^{\circ})\approx0.573576 \)
\( GI = 5.83095\times0.573576\approx3.3 \)

Answer:

\( 3.3 \)