Y is 60 km away from X on a bearing of $135^{\circ}$. Z is 80 km away from X on a bearing of $225^{\circ}$. We need to find: (a) the distance of Z from Y; (b) the bearing of Z from Y.

1. Problem Description

Y is 60 km away from X on a bearing of $135^{\circ}$. Z is 80 km away from X on a bearing of $225^{\circ}$. We need to find:

(a) the distance of Z from Y;

(b) the bearing of Z from Y.

2. Solution Steps

(a) To find the distance of Z from Y, we can use the cosine rule. First, we need to find the angle YXZ.

The bearing of Y from X is $135^{\circ}$ and the bearing of Z from X is $225^{\circ}$.

The angle YXZ is the difference between the two bearings:

$YXZ = 225^{\circ} - 135^{\circ} = 90^{\circ}$.

Now, we can use the cosine rule to find the distance YZ:

$YZ^2 = XY^2 + XZ^2 - 2(XY)(XZ)cos(YXZ)$

$YZ^2 = 60^2 + 80^2 - 2(60)(80)cos(90^{\circ})$

Since $cos(90^{\circ}) = 0$, the equation simplifies to:

$YZ^2 = 60^2 + 80^2$

$YZ^2 = 3600 + 6400$

$YZ^2 = 10000$

$YZ = \sqrt{10000}$

$YZ = 100$ km

(b) To find the bearing of Z from Y, we first need to find the angle XYZ.

Since triangle XYZ is a right-angled triangle, we can use trigonometric ratios.

$tan(XYZ) = \frac{XZ}{XY} = \frac{80}{60} = \frac{4}{3}$

$XYZ = arctan(\frac{4}{3}) \approx 53.13^{\circ}$

Now, we need to find the angle between the North direction at Y and the line YZ.

The bearing of Y from X is $135^{\circ}$, which means that the angle between the North direction at X and the line XY is $135^{\circ}$. The angle between the East direction at X and the line XY is $135^{\circ} - 90^{\circ} = 45^{\circ}$. This also means the angle between South at X and the line XY is $45^{\circ}$.

The angle between the North direction at Y and the line YX is $180 + 135 = 315$. However, we need the "back bearing" which is $135^{\circ} + 180^{\circ} = 315^{\circ}$.

The back bearing is the same as $315 - 360 = -45$, which can be thought of as $180 + 135 = 315$ degrees or $360 - 45 = 315$ degrees. Then the angle from North at Y to XY is $315$.

Therefore the angle from the East from Y to XY is $315 - 90 = 225$.

Since angle XYZ is $53.13^{\circ}$, the angle between North and the line YZ can be calculated as follows.

Angle between South and line YX = $45^{\circ}$. Then the bearing of Z from Y can be $180 + XYZ = 180 + 53.13 = 233.13$.

The bearing of Z from Y = $135 + 180 +53.13 \approx 180+53.13$ or $225-90-atan(\frac{3}{4}) = 225-90-36.87 = 98.13$ wrong. The value has to be between 180 and

2

7

0. $180+arctan(8/6)$

$\arctan(4/3) \approx 53.13^{\circ}$

Bearing = $135 + 180^{\circ} - 53.13^{\circ}=315 - 53.13=261.87$

3. Final Answer

(a) The distance of Z from Y is 100 km.

(b) The bearing of Z from Y is approximately $261.9^{\circ}$.