Question

find su.
write your answer in simplified, rationalized form. do not round.
su =

Explanation:

Step1: Identify triangle type and trigonometric ratio

This is a right - triangle \( \triangle STU \) with \( \angle S = 90^{\circ} \), \( \angle T=60^{\circ} \), and \( ST = 4\sqrt{10} \). We want to find \( SU \). We can use the tangent function, which is defined as \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \) in a right - triangle. For \( \angle T = 60^{\circ} \), the side opposite to \( \angle T \) is \( SU \) and the side adjacent to \( \angle T \) is \( ST \). So, \( \tan(60^{\circ})=\frac{SU}{ST} \).

Step2: Recall the value of \( \tan(60^{\circ}) \) and substitute the value of \( ST \)

We know that \( \tan(60^{\circ})=\sqrt{3} \) and \( ST = 4\sqrt{10} \). Substituting these values into the tangent formula: \( \sqrt{3}=\frac{SU}{4\sqrt{10}} \).

Step3: Solve for \( SU \)

To solve for \( SU \), we multiply both sides of the equation by \( 4\sqrt{10} \). So, \( SU = 4\sqrt{10}\times\sqrt{3} \). Using the property of square roots \( \sqrt{a}\times\sqrt{b}=\sqrt{ab} \), we get \( SU = 4\sqrt{30} \).

Answer:

\( 4\sqrt{30} \)