To find the surface area of the triangular prism, we need to calculate the area of the two triangular faces and the three rectangular faces, then sum them.

Step 1: Area of the triangular face

The given triangle is a right triangle with legs 9 cm and 12 cm.

$\text{Area of the triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 9 \, \text{cm} \times 12 \, \text{cm} = 54 \, \text{cm}^2$

Since there are two triangular faces, their total area is: $2 \times 54 \, \text{cm}^2 = 108 \, \text{cm}^2$

Step 2: Area of the three rectangular faces

The rectangular faces have the following dimensions:

1. One rectangle with dimensions $15 \, \text{cm} \times 5 \, \text{cm}$ (the length of the prism and the height of the prism): $\text{Area} = 15 \, \text{cm} \times 5 \, \text{cm} = 75 \, \text{cm}^2$

2. One rectangle with dimensions $9 \, \text{cm} \times 5 \, \text{cm}$ (height of the triangular face and length of the prism): $\text{Area} = 9 \, \text{cm} \times 5 \, \text{cm} = 45 \, \text{cm}^2$

3. One rectangle with dimensions $12 \, \text{cm} \times 5 \, \text{cm}$ (base of the triangular face and length of the prism): $\text{Area} = 12 \, \text{cm} \times 5 \, \text{cm} = 60 \, \text{cm}^2$

Step 3: Total surface area

Now, sum all the areas:

$\text{Total Surface Area} = 108 \, \text{cm}^2 + 75 \, \text{cm}^2 + 45 \, \text{cm}^2 + 60 \, \text{cm}^2 = 288 \, \text{cm}^2$

Thus, the surface area of the triangular prism is 288 cm².

Would you like a deeper breakdown of any step or have any more questions?

Here are 5 related questions to consider:

1. How would the surface area change if the height of the prism increased?
2. What is the volume of this triangular prism?
3. How can we calculate the surface area of a non-right triangular prism?
4. How does the surface area formula apply to different polygonal prisms?
5. How do you find the slant height in oblique triangular prisms?

Tip: When calculating surface area, always check if all faces are accounted for (including opposite and similar faces).