Question

solve the right triangle.

triangle with right angle at h, hi = 6, gh = 6√3, vertices g, h, i

write your answers in simplified, rationalized form. do not round.

gi =

m∠i = °

m∠g = °

Explanation:

Step1: Find the hypotenuse \( GI \) using Pythagorean theorem

In right triangle \( GHI \), \( \angle H = 90^\circ \), \( GH = 6\sqrt{3} \), \( HI = 6 \). By Pythagorean theorem \( GI^2 = GH^2 + HI^2 \).
\[

$$\begin{align*} GI^2&=(6\sqrt{3})^2 + 6^2\\ &= 6^2\times(\sqrt{3})^2+ 6^2\\ &= 36\times3 + 36\\ &= 108 + 36\\ &= 144 \end{align*}$$

\]
So \( GI=\sqrt{144} = 12 \).

Step2: Find \( m\angle I \) using trigonometric ratio

\( \tan(\angle I)=\frac{GH}{HI}=\frac{6\sqrt{3}}{6}=\sqrt{3} \). We know that \( \tan(60^\circ)=\sqrt{3} \), so \( m\angle I = 60^\circ \).

Step3: Find \( m\angle G \)

In a right triangle, the sum of acute angles is \( 90^\circ \). So \( m\angle G=90^\circ - m\angle I = 90^\circ - 60^\circ = 30^\circ \).

Answer:

\( GI = 12 \)
\( m\angle I = 60^\circ \)
\( m\angle G = 30^\circ \)