Question

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

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Answer 1

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

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}`.