Question

solve the right triangle.
write your answers in simplified, rationalized form. do not round.
fg=

m∠g=

m∠f=

Explanation:

Step1: Find the length of \(FG\) using the Pythagorean theorem

In a right - triangle \(EFG\) with \(\angle E = 90^{\circ}\), by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 6\), \(b = 6\sqrt{3}\), and \(c=FG\).

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Step2: Find \(m\angle G\) using the tangent function

\(\tan G=\frac{EF}{EG}\). Here, \(EF = 6\) and \(EG=6\sqrt{3}\).

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Since \(\tan30^{\circ}=\frac{1}{\sqrt{3}}\), \(m\angle G = 30^{\circ}\)

Step3: Find \(m\angle F\) using the angle - sum property of a triangle

In \(\triangle EFG\), \(m\angle E+m\angle F+m\angle G = 180^{\circ}\). Since \(m\angle E = 90^{\circ}\) and \(m\angle G=30^{\circ}\)

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Answer:

\(FG = 12\)
\(m\angle G=30\)
\(m\angle F = 60\)