Point Y is 60 km away from point X on a bearing of 135 degrees. Point Z is 80 km away from point X on a bearing of 225 degrees. We need to find the distance of Z from Y and the bearing of Z from Y.

1. Problem Description

Point Y is 60 km away from point X on a bearing of 135 degrees. Point Z is 80 km away from point X on a bearing of 225 degrees. We need to find the distance of Z from Y and the bearing of Z from Y.

2. Solution Steps

First, let's find the angle $YXZ$. Since the bearing of Y from X is 135 degrees and the bearing of Z from X is 225 degrees, the angle $YXZ$ is:

$YXZ = 225 - 135 = 90$ degrees.

Now, we have a triangle $XYZ$ with $XY = 60$ km, $XZ = 80$ km, and the angle $YXZ = 90$ degrees. We can use the Pythagorean theorem to find the distance $YZ$:

$YZ^2 = XY^2 + XZ^2$

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

$YZ^2 = 3600 + 6400$

$YZ^2 = 10000$

$YZ = \sqrt{10000}$

$YZ = 100$ km.

So, the distance of Z from Y is 100 km.

Now we need to find the bearing of Z from Y. First, let's find the angle $XY Z$.

Let $\theta = \angle XYZ$.

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

$\theta = \arctan(\frac{4}{3}) \approx 53.13$ degrees.

The bearing of Y from X is 135 degrees. We want the bearing of Z from Y.

The bearing of X from Y is $135 + 180 = 315$ degrees (or $135-180 = -45$ degrees, which is equivalent to 315 degrees).

Now, the angle between North at Y and the line YX is 315 degrees.

The angle $XYZ = 53.13$ degrees.

The angle between North at Y and YZ is $315 + 53.13 - 360$.

If the result is negative, we add 360 to it.

Bearing of Z from Y $= 315 - 53.13$

Bearing of Z from Y $= 261.87$ degrees.

2. Final Answer

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

(b) The bearing of Z from Y is approximately 261.87 degrees.