Question

solve the right triangle.

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

m∠r =

m∠s =

Explanation:

Step1: Find hypotenuse \( RS \)

Use Pythagorean theorem: \( RS^2 = RQ^2 + QS^2 \). Substitute \( RQ = 3\sqrt{3} \), \( QS = 3 \):
\( RS^2 = (3\sqrt{3})^2 + 3^2 = 27 + 9 = 36 \). Thus, \( RS = \sqrt{36} = 6 \).

Step2: Find \( m\angle R \)

Use tangent: \( \tan(\angle R) = \frac{QS}{RQ} = \frac{3}{3\sqrt{3}} = \frac{1}{\sqrt{3}} \). \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so \( m\angle R = 30^\circ \).

Step3: Find \( m\angle S \)

In right triangle, angles sum to \( 180^\circ \). So \( m\angle S = 90^\circ - 30^\circ = 60^\circ \).

Answer:

\( RS = 6 \)
\( m\angle R = 30 \)
\( m\angle S = 60 \)