Question

The bearing of two points x and y from z are 45° and 135° respectively . if |zx|=8cm and |zy|=6cm, find |xy|.

Answer 1

Answer:

$|{\sf XY}| = 10\; {\rm cm}$.

Step-by-step explanation:

Refer to the diagram attached. The dashed segment attached to $\!{\sf Z}$ points to the north. Rotating this segment clockwise with point ${\sf Z}\!\!$ as the fixed center of rotation would eventually align this segment with the one between point $\!\!{\sf Z}$ and point $\!\!{\sf X}$. The bearing of point ${\sf X}$ from point ${\sf Z}$ is the size of the angle between these two line segments when measured in the clockwise direction.

Subtract the bearing of ${\sf Y}$ from ${\sf Z}$ from the bearing of ${\sf X}$ from ${\sf Z}$ to find the measure of the angle $\angle {\sf YZX}$:

$\begin{aligned}\angle {\sf YZX} &= 135^{\circ} - 45^{\circ} \\ &= 90^{\circ}\end{aligned}$.

Thus, triangle $\triangle {\sf YZX}$ is a right triangle ($90^{\circ}$) with segment ${\sf YX}$ as the hypotenuse. It is given that $|{\sf XZ}| = 6\; {\rm cm}$ whereas $|{\sf ZY}| = 6\; {\rm cm}$. Thus, by Pythagorean's Theorem:

$\begin{aligned}|{\sf ZY}| &= \sqrt{|{\sf ZX}|^{2} + |{\sf ZY}|^{2}} \\ &= \sqrt{(8\; {\rm cm})^{2} + (6\; {\rm cm})^{2}} \\ &= 10\; {\rm cm}\end{aligned}$.