Question

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

m∠e=

m∠f=

Explanation:

Step1: Find \(EF\) using Pythagorean theorem

In right - triangle \(DEF\), by Pythagorean theorem \(EF=\sqrt{FD^{2}+ED^{2}}\).
Given \(FD = 4\sqrt{6}\) and \(ED=12\sqrt{2}\).

$$
LATEXBLOCK0
$$

Step2: Find \(m\angle E\) using sine function

\(\sin E=\frac{FD}{EF}\).
Since \(FD = 4\sqrt{6}\) and \(EF = 8\sqrt{6}\), then \(\sin E=\frac{4\sqrt{6}}{8\sqrt{6}}=\frac{1}{2}\).
So \(m\angle E = 30^{\circ}\)

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

In \(\triangle DEF\), \(m\angle D=90^{\circ}\), \(m\angle E + m\angle F+m\angle D=180^{\circ}\).
Since \(m\angle E = 30^{\circ}\) and \(m\angle D = 90^{\circ}\), then \(m\angle F=180-(90 + 30)=60^{\circ}\)

Answer:

\(EF = 8\sqrt{6}\), \(m\angle E=30^{\circ}\), \(m\angle F = 60^{\circ}\)