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The problem asks to find the perimeter of triangle \( \triangle XYZ \), which is depicted on the right side of the image.

### Information from the triangle:
- Angle at \( Y = 72^\circ \)
- Angle at \( Z = 63^\circ \)
- Angle at \( X = 45^\circ \)
- Side \( XY = 33 \)
- Side \( YZ = 25.5 \)

To find the perimeter of \( \triangle XYZ \), we need the length of side \( XZ \). We can use the **Law of Cosines** to calculate \( XZ \).

The Law of Cosines states:

\[
XZ^2 = XY^2 + YZ^2 - 2 \cdot XY \cdot YZ \cdot \cos(\angle Y)
\]

### Substituting the known values:
- \( XY = 33 \)
- \( YZ = 25.5 \)
- \( \angle Y = 72^\circ \)

Now let's calculate \( XZ \).The length of side \( XZ \) is approximately \( 34.9 \) units.

### Now, to find the perimeter of \( \triangle XYZ \), we add up the lengths of all three sides:

\[
\text{Perimeter} = XY + YZ + XZ = 33 + 25.5 + 34.9 = 93.4
\]

Thus, the perimeter of \( \triangle XYZ \) is approximately **93.4 units**.

Would you like further details on any part of this process or help with related concepts?

Here are five related questions you might find useful:
1. How is the Law of Cosines derived?
2. How can the Law of Sines be applied to this triangle?
3. Can the perimeter be found using a different method, such as the Law of Sines?
4. What is the difference between the Law of Cosines and the Law of Sines?
5. How would you calculate the area of \( \triangle XYZ \)?

**Tip**: Always ensure the angles you use are in degrees or radians, matching the mode of your calculator or function.

Find the perimeter of triangle XYZ. Round your answer to the nearest tenth if necessary. Figures are not necessarily drawn to scale.

Learn how to find the perimeter of triangle XYZ using the Law of Cosines. This problem involves calculating side lengths based on given angles and known sides, and applying the Law of Cosines to solve for the missing side. Suitable for students in Grades 10-12.

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