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Let's work through the given problem step by step:

### Problem Overview:
- Triangle \( XYZ \) has sides \( XY = 6 \, \text{cm} \) and \( ZX = 4 \, \text{cm} \).
- Part of the shape is a circular arc \( ZW \), which is part of a circle centered at \( X \) with a radius of \( 4 \, \text{cm} \).
- The point \( W \) lies on the line \( XY \), and the angle \( ZXW = \alpha \, \text{radians} \).
- The length of the arc \( ZW \) is given as \( \frac{40\pi}{9} \, \text{cm} \).

You are asked to find:
1. **The exact value of \( \alpha \) in radians**.
2. **The area of triangle \( XYZ \)**.

### Part (a): Finding \( \alpha \)
We know the formula for the length of an arc is:
\[
\text{Arc length} = r \times \theta
\]
Where \( r \) is the radius and \( \theta \) is the angle subtended by the arc at the center (in radians).

Here:
- The radius \( r = 4 \, \text{cm} \),
- The arc length is \( \frac{40\pi}{9} \, \text{cm} \),
- The angle subtended is \( \alpha \).

So, we can set up the equation:
\[
\frac{40\pi}{9} = 4 \times \alpha
\]
Solving for \( \alpha \):
\[
\alpha = \frac{40\pi}{9 \times 4} = \frac{10\pi}{9} \, \text{radians}
\]

Thus, the exact value of \( \alpha \) is \( \frac{10\pi}{9} \, \text{radians} \).

### Part (b): Area of Triangle \( XYZ \)
To find the area of triangle \( XYZ \), we can use the formula for the area of a triangle given two sides and the included angle:
\[
\text{Area} = \frac{1}{2}ab\sin(C)
\]
Where:
- \( a = 6 \, \text{cm} \) (side \( XY \)),
- \( b = 4 \, \text{cm} \) (side \( ZX \)),
- \( C = \alpha = \frac{10\pi}{9} \, \text{radians} \).

Now, let's compute the area using this formula.

It seems the computed area is negative, which typically means that the sine of the angle has given a negative value. Since angles larger than \( \pi \) radians (180°) result in a negative sine value, it suggests the angle \( \alpha = \frac{10\pi}{9} \, \text{radians} \) is larger than \( \pi \).

Since we need the positive area, we should correct for this by considering the positive value of \( \sin(\alpha) \).

Thus, the **area of triangle \( XYZ \)** is approximately **4.1 cm²**.

Would you like to go over any of the steps in more detail, or do you have other questions? 

Here are five related questions for practice:
1. How would the area formula change if you were given all three sides of the triangle?
2. How can you convert the angle \( \alpha \) from radians to degrees?
3. What would the length of the arc be if the radius were doubled?
4. Can the sine rule be used to find unknown angles in this triangle?
5. What would be the perimeter of the triangle \( XYZ \)?

**Tip:** Always check the range of an angle when calculating trigonometric functions like sine to ensure you're interpreting the result correctly.

The shape comprises a triangle XYZ, with XY = 6cm and ZX = 4cm, and part of a circle. The circular arc ZW is a major arc of the circle with centre X and radius 4cm. The point W lies on the line XY and the angle ZXW = α radians. Given that the exact length of arc ZW is 40π/9 cm, find:

a) the exact value of α in radians
b) the area of triangle XYZ, giving your answer to 1 decimal place.

Learn how to calculate the angle in radians and the area of a triangle in a problem involving arc length and trigonometric principles. This question features geometry concepts and is suitable for students in Grades 10-12.

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