To solve the problem based on the net of the triangular prism:

a) Work out the length $x$:

The green face is one of the rectangular faces of the prism, and its width is $x$, which corresponds to the base of the triangular face of the prism.

The triangular face is a right triangle with sides of 3 cm and 4 cm (these are the legs of the triangle). To find the base $x$ (the hypotenuse of this triangle), we use the Pythagorean theorem:

$x^2 = 3^2 + 4^2$ $x^2 = 9 + 16 = 25$ $x = \sqrt{25} = 5 \, \text{cm}$

So, the length of $x$ is 5 cm.

b) Work out the area of the shaded face:

The shaded face is a rectangle with a width of $x = 5 \, \text{cm}$ and a height of 9 cm. The area of a rectangle is calculated using the formula:

$\text{Area} = \text{width} \times \text{height}$ $\text{Area} = 5 \, \text{cm} \times 9 \, \text{cm} = 45 \, \text{cm}^2$

So, the area of the shaded face is 45 cm².

Would you like further details on any part of the solution?

Here are five related questions you might want to consider:

1. How would you calculate the surface area of the entire triangular prism?
2. What is the volume of this triangular prism?
3. If the dimensions of the triangle change, how does it affect the area of the shaded face?
4. How do you find the area of other triangular faces of the prism?
5. How would you use the Pythagorean theorem in 3D geometry?

Tip: The Pythagorean theorem is extremely useful in geometry problems involving right triangles, especially when dealing with 3D shapes like prisms.