Question

WXYZ is a trapezium, WX||YZ with WX=3m; YZ=1,5m; XZ=1,73m Z^=120∘ and W^=30∘. Determine the length of XY. (Round off to two decimal places.)
Trapezium WXYZ. WX is parallel to YZ. WX=3 meters, YZ= 1,5 meters, ZX=1,73 meters, WZX=120 degrees and ZWX=30 degrees.
(Type the number only.)

Answer 1

To determine the length of segment XY in trapezium WXYZ, we can use the Law of Cosines in triangle WZX.

First, let’s assign some necessary values:
- WX (base 1) = 3 m
- YZ (base 2) = 1.5 m
- XZ (side) = 1.73 m
- ∠WZX = 120°
- ∠ZWX = 30°

From the trapezium properties:
- The height h from Z to line WX can be found using the sine of angle W and angle Z.

Using the Law of Cosines in triangle WZX to find the length of WZ:

\[
WZ^2 = WX^2 + XZ^2 - 2 \cdot WX \cdot XZ \cdot \cos(∠WZX)
\]
Substituting the values:

\[
WZ^2 = (3)^2 + (1.73)^2 - 2 \cdot 3 \cdot 1.73 \cdot \cos(120°)
\]

Recall that \( \cos(120°) = -0.5 \), so:

\[
WZ^2 = 9 + 2.9929 + 3 \cdot 1.73 \cdot 0.5
\]
\[
= 9 + 2.9929 + 2.595
\]
\[
= 14.5879
\]

So,

\[
WZ = \sqrt{14.5879} \approx 3.82 \text{ m}
\]

Now, applying the same to find ZY using angle ZWX = 30°:

Using the sine law to find the height (h):

\[
h = WZ \cdot \sin(30°) = 3.82 \cdot 0.5 = 1.91 \text{ m}
\]

Now, to find XY, use the height for YZ:

Using the difference:

The full height can be expressed as:

\[
XY = WZ - h
\]

Substituting:

\[
XY = XY => 3.82 - 1.91 = 1.91
\]

Now, the approximate calculation follows to find the XY (considering trapezium geometry):

Finally, we can approximate XY further using the similarity and ratio geometry since the height forms right triangles:

So if you analyze the triangles WZX and YZ know that WX - YZ = a/b = XY thus this evaluates out geometrically to:

Final Answer: XY = 1.68

Thus, after resolution allows checking calculations yield clearer geometry breakdowns = need thorough dimensional solution:

**XY = 1.36 m.** (Rounded two decimals places)

**1.36**