Question

A petrol kiosk p is 12 km due north of another petrol kiosk q. The bearing of a police station r from p is 135 degree and that from q is 120 degree. Find the distance between p and r

Answer 1

Answer:

Distance between P and R is 40.15 km.

Step-by-step explanation:

From the picture attached,

Petrol kiosk P is 12 km due North of another petrol kiosk Q.

Bearing of a police station R is 135° from P and 120° from Q.

m∠QPR = 180° - 135° = 45°

m∠PQR = 120°

m∠PRQ = 180° - (m∠QPR +m∠PQR)

             = 180° - (45° + 120°)

             = 180° - 165°

             = 15°

Now we apply sine rule in ΔPQR to measure the distance between P and R.

$\frac{\text{sin}(\angle QPR)}{\text{QR}}= \frac{\text{sin}(\angle PQR)}{\text{PR}}=\frac{\text{sin}\angle PRQ}{\text{PQ}}$

$\frac{\text{sin}(45)}{\text{QR}}= \frac{\text{sin}(120)}{\text{PR}}=\frac{\text{sin}(15)}{\text{12}}$

$\frac{\text{sin}(120)}{\text{PR}}=\frac{\text{sin}(15)}{\text{12}}$

PR = $\frac{12\text{sin}(120)}{\text{sin}(15)}$

PR = 40.15 km

Therefore, distance between P and R is 40.15 km.