Question

solve the right triangle.
write your answers as integers or as decimals rounded to the nearest tenth.
pq =

m∠r =

m∠q =

Explanation:

Step1: Find \( PQ \) using Pythagorean theorem

In right - triangle \( \triangle RPQ \), by the Pythagorean theorem \( PQ=\sqrt{QR^{2}-RP^{2}} \). Given \( QR = 5.4 \) and \( RP=2 \), then \( PQ=\sqrt{5.4^{2}-2^{2}}=\sqrt{29.16 - 4}=\sqrt{25.16}\approx5.0 \).

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

\(\sin Q=\frac{RP}{QR}\). Substitute \( RP = 2 \) and \( QR=5.4 \), so \(\sin Q=\frac{2}{5.4}\approx0.3704\). Then \( Q=\sin^{-1}(0.3704)\approx21.8^{\circ}\).

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

Since the sum of angles in a triangle is \( 180^{\circ} \) and in right - triangle \( \angle P = 90^{\circ} \), \( m\angle R=90^{\circ}-m\angle Q \). Substitute \( m\angle Q\approx21.8^{\circ} \), so \( m\angle R = 90^{\circ}-21.8^{\circ}=68.2^{\circ}\).

Answer:

\( PQ\approx5.0 \), \( m\angle Q\approx21.8^{\circ} \), \( m\angle R\approx68.2^{\circ} \)