Question

find gi.
h to i is 18√22, angle at i is 30°, angle at g is right angle. write your answer in simplified, rationalized form. do not round.
gi =

Explanation:

Step1: Identify triangle type

Triangle \( \triangle GHI \) is right - angled at \( G \), with \( \angle I = 30^\circ \) and hypotenuse \( HI=18\sqrt{22} \). In a 30 - 60 - 90 right triangle, the side adjacent to the \( 30^\circ \) angle (here \( GI \)) is related to the hypotenuse by the cosine function. The cosine of an angle \( \theta \) in a right triangle is defined as \( \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} \).

Step2: Apply cosine formula

We know that \( \cos(30^\circ)=\frac{GI}{HI} \). We know that \( \cos(30^\circ)=\frac{\sqrt{3}}{2} \) and \( HI = 18\sqrt{22} \). So we can solve for \( GI \) by rearranging the formula: \( GI=HI\times\cos(30^\circ) \).
Substitute the values: \( GI = 18\sqrt{22}\times\frac{\sqrt{3}}{2} \).
Simplify the expression: \( GI=\frac{18}{2}\times\sqrt{22\times3}=9\sqrt{66} \).

Answer:

\( 9\sqrt{66} \)