https://file.aiquickdraw.com/mathbot/file/1726652034590y97w2sue.jpg

We are tasked with finding the length of side \( WZ \) in the trapezium \( WXYZ \).

From the diagram, we can see the following:
- \( XY = 8 \, \text{cm} \)
- \( YZ = 6 \, \text{cm} \) (vertical height)
- \( XW = 12.9 \, \text{cm} \)
- \( WZ \) is the missing side we need to find, and it is horizontal.

### Step-by-step approach:

1. Since \( WZ \) is parallel to \( XY \) and forms a right triangle with \( YZ \) as the height and the horizontal distance between \( XW \) and \( XY \), we can apply the Pythagorean theorem in the right triangle \( YZW \).

2. The horizontal distance between \( W \) and \( Y \) can be calculated as:
   \[
   XW - XY = 12.9 \, \text{cm} - 8 \, \text{cm} = 4.9 \, \text{cm}
   \]
   This forms the base of the right triangle \( YZW \).

3. Now apply the Pythagorean theorem:
   \[
   WZ^2 = YZ^2 + (XW - XY)^2
   \]
   Substituting the values:
   \[
   WZ^2 = 6^2 + 4.9^2
   \]
   \[
   WZ^2 = 36 + 24.01 = 60.01
   \]
   \[
   WZ = \sqrt{60.01} \approx 7.75 \, \text{cm}
   \]

Thus, the length of \( WZ \) is approximately **7.75 cm**.

Would you like further details or have any questions?

Here are some related questions for you to consider:
1. How do we calculate the area of trapezium \( WXYZ \)?
2. What other properties of trapeziums can we explore from this diagram?
3. How would the problem change if \( YZ \) was not perpendicular to \( XW \)?
4. Can we find the perimeter of the trapezium using the given information?
5. How does the Pythagorean theorem apply in different contexts of geometry?

**Tip:** When solving geometry problems involving right angles, always check if the Pythagorean theorem applies, as it simplifies finding unknown lengths.

Find the length of WZ in trapezium WXYZ, where XY = 8 cm, YZ = 6 cm, XW = 12.9 cm.

In this geometry problem, find the length of side WZ in trapezium WXYZ using the Pythagorean theorem. With sides XY = 8 cm, YZ = 6 cm, and XW = 12.9 cm, follow a step-by-step solution to calculate WZ. Suitable for students in Grades 8-10.

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