15 In \( \triangle P Q R, P Q Q R=84^{\circ}, Q \widehat{P} R=43^{\circ} \) and \( I P Q /=5 \mathrm{~cm} \). Find \( / Q R / \) in cm, correct to 1 decimal place. (WASSCE)

To find \( QR \) in triangle \( PQR \) with the given angles and side length, we can use the Law of Sines. First, we find the missing angle \( R \) using the fact that the sum of angles in a triangle is \( 180^\circ \). \[ R = 180^\circ - P - Q = 180^\circ - 84^\circ - 43^\circ = 53^\circ \] Now that we have all the angles: - \( \angle P = 84^\circ \) - \( \angle Q = 43^\circ \) - \( \angle R = 53^\circ \) Using the Law of Sines: \[ \frac{PQ}{\sin R} = \frac{QR}{\sin P} \] Substituting the known values: \[ \frac{5 \text{ cm}}{\sin(53^\circ)} = \frac{QR}{\sin(84^\circ)} \] To find \( QR \): \[ QR = \frac{5 \cdot \sin(84^\circ)}{\sin(53^\circ)} \] Calculating the sine values: - \( \sin(84^\circ) \approx 0.998 \) - \( \sin(53^\circ) \approx 0.798 \) Substituting these values: \[ QR \approx \frac{5 \cdot 0.998}{0.798} \approx \frac{4.99}{0.798} \approx 6.24 \text{ cm} \] Therefore, \( QR \approx 6.2 \text{ cm} \) when rounded to 1 decimal place.