Questions: Find the exact value of tan F in simplest form.

Find the exact value of \(\tan F\) in simplest form.

Identify the sides of the triangle relative to angle F.

EF is the opposite side to angle F. DF is the hypotenuse of triangle DEF. FD is the adjacent side to angle F.

Calculate the length of side EF.

Triangle DEF is a right triangle, so we can use the Pythagorean theorem: \(DE^2 + EF^2 = DF^2\) \(DE^2 + EF^2 = (\sqrt{77})^2\) \(DE = 2\). \(DF = 9\) So \(2^2 + EF^2 = 77\) \(4 + EF^2 = 77\) \(EF^2 = 77 - 4\) \(EF^2 = 73\) \(EF = \sqrt{73}\)

Calculate the tangent of angle F.

\(\tan F = \frac{opposite}{adjacent} = \frac{DE}{EF} = \frac{2}{\sqrt{73}}\)

Rationalize the denominator.

Multiply the numerator and denominator by \(\sqrt{73}\): \(\tan F = \frac{2}{\sqrt{73}} \times \frac{\sqrt{73}}{\sqrt{73}} = \frac{2\sqrt{73}}{73}\)

\(\boxed{\tan F = \frac{2\sqrt{73}}{73}}\)

\(\tan F = \frac{2\sqrt{73}}{73}\)