Question

find r.
write your answer as an integer or as a decimal rounded to the nearest tenth.
r =
submit

Explanation:

Step1: Identify triangle type and angles

Triangle \( PQR \) is right - angled at \( Q \) (since \( \angle Q = 90^{\circ} \), as it is a right - triangle with \( PQ=r \), \( QR = 5 \), and \( \angle P=29^{\circ} \), \( \angle R = 64^{\circ} \)). We know that in a right - triangle, \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \). For \( \angle P = 29^{\circ} \), the opposite side to \( \angle P \) is \( QR = 5 \) and the adjacent side is \( PQ=r \).

Step2: Apply the tangent formula

We use the tangent function: \( \tan(\angle P)=\frac{QR}{PQ} \). Substituting \( \angle P = 29^{\circ} \), \( QR = 5 \) and \( PQ=r \), we get \( \tan(29^{\circ})=\frac{5}{r} \).

Step3: Solve for \( r \)

We can re - arrange the formula to solve for \( r \): \( r=\frac{5}{\tan(29^{\circ})} \). We know that \( \tan(29^{\circ})\approx0.5543 \). Then \( r=\frac{5}{0.5543}\approx9.0 \) (rounded to the nearest tenth).

Answer:

\( 9.0 \)