Question

find gi.

triangle image with right angle at i, angle at h is 64 degrees, hypotenuse gh is 10

write your answer as an integer or as a decimal rounded to the nearest tenth.
gi =
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Explanation:

Step1: Identify the trigonometric ratio

In right triangle \( \triangle GIH \), we know the hypotenuse \( GH = 10 \) and the angle at \( H \) is \( 64^\circ \). We need to find the length of \( GI \), which is the side opposite to the angle \( 64^\circ \). The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse. So, \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \). Here, \( \theta = 64^\circ \), opposite side is \( GI \), and hypotenuse is \( GH = 10 \). So, \( \sin(64^\circ)=\frac{GI}{10} \).

Step2: Solve for \( GI \)

To find \( GI \), we can rearrange the formula: \( GI = 10\times\sin(64^\circ) \). Now, we calculate \( \sin(64^\circ) \). Using a calculator, \( \sin(64^\circ)\approx0.8988 \). Then, \( GI = 10\times0.8988 = 8.988 \approx 9.0 \) (rounded to the nearest tenth).

Answer:

\( 9.0 \)