2. Emefa walked \( 2 t \mathrm{~km} \) from a village, P to \( v_{i_{\text {sit }}} \) a friend in another village, Q on a bearing of \( 065^{\circ} \). After spending some time with her friend, she continued to a nearby town, R , \( 3 t \mathrm{~km} \) away on a bearing of \( 155^{\circ} \). If the distance between P and R is \( 6 \sqrt{13} \mathrm{~km} \) : (a) illustrate the information in a diagram; (b) calculate, correct to the nearest whole number, the: (i) value of \( t \); (ii) bearing of R from P .

To illustrate Emefa's journey, imagine a coordinate plane where you can plot points denoting her path: 1. Start at point P (Village P). 2. From P, draw a line at a 65° angle from North, and mark a point for Q (Village Q) at a distance of \(2t\) km. 3. From Q, draw a line at a 155° angle from North towards point R (Town R), marking this at a distance of \(3t\) km. Now, to solve for \(t\): Using the Law of Cosines on triangle PQR where PR = \(6\sqrt{13}\): \[ PR^2 = PQ^2 + QR^2 - 2(PQ)(QR)\cos(\angle QPR) \] Calculate the angle QPR using bearings: \(\angle QPR = 155° - 65° = 90°\). Hence, cos(90°) = 0. So rearranging gives: \[ (6\sqrt{13})^2 = (2t)^2 + (3t)^2 \] \[ 468 = 4t^2 + 9t^2 \] \[ 468 = 13t^2 \Rightarrow t^2 = \frac{468}{13} \Rightarrow t = \sqrt{36} = 6. \] Now, for part (ii), calculating the bearing of R from P involves finding the angles to add: 1. From P to Q: \(65^\circ\) 2. From Q to R: \(155^\circ - 180^\circ = -25^\circ\). Thus, the concept of bearings implies an adjustment considering clockwise from North. Now, summing these gives: \[ \text{Bearing of R from P} = 65 + 130 (from east 155°) = 195°. \] So, the bearing of R from P is \(195^\circ\). --- Fun Fact: The Law of Cosines, or if you like, the “Triangle’s Secret Sauce”, is used for solving those tricky triangle problems where you don’t have a right angle to work with, often stealing the spotlight in geometry class! Further Reading: If you're intrigued by bearings and triangle properties, try delving into trigonometry textbooks or online courses that cover navigation and surveying techniques—they're like treasure maps for the mathematically adventurous!