To solve this problem, I will analyze the triangular prism's dimensions and calculate its edge lengths and surface area. Here's the step-by-step solution:

1. Extract the dimensions:

From the image:

• The rectangular face has a width of 9.5 cm and a length of 15 cm.
• The triangular face has a base of 10 cm (matching the prism's width).

The triangular prism consists of:

• 2 triangular faces (identical)
• 3 rectangular faces

2. Find missing information (height of the triangle):

To calculate the surface area, we need the height of the triangle in the triangular faces. For a right triangle with:

• Hypotenuse = 15 cm
• Base = 10 cm

We use the Pythagorean theorem to find the height $h$: $h^2 = \text{Hypotenuse}^2 - \text{Base}^2$ $h^2 = 15^2 - 10^2$ $h^2 = 225 - 100 = 125$ $h = \sqrt{125} \approx 11.18 \, \text{cm}$

3. Calculate surface area:

The surface area of the triangular prism is the sum of:

1. Area of 2 triangular faces: $\text{Area of triangle} = \frac{1}{2} \times \text{Base} \times \text{Height}$ $\text{Area} = \frac{1}{2} \times 10 \times 11.18 = 55.9 \, \text{cm}^2$ $\text{Total for 2 triangles} = 2 \times 55.9 = 111.8 \, \text{cm}^2$

2. Area of 3 rectangular faces:

   • Two rectangles with dimensions $9.5 \, \text{cm} \times 15 \, \text{cm}$: $\text{Area} = 9.5 \times 15 = 142.5 \, \text{cm}^2 \quad (\text{each rectangle})$ $\text{Total for 2 rectangles} = 2 \times 142.5 = 285 \, \text{cm}^2$
   • One rectangle with dimensions $10 \, \text{cm} \times 15 \, \text{cm}$: $\text{Area} = 10 \times 15 = 150 \, \text{cm}^2$

3. Add all areas: $\text{Total Surface Area} = 111.8 + 285 + 150 = 546.8 \, \text{cm}^2$

4. Edge lengths:

A triangular prism has:

• 3 edges on each triangular face (base = 10 cm, height edge ≈ 11.18 cm, hypotenuse = 15 cm)
• 3 rectangular edges (length = 15 cm)

Thus, the edge lengths are:

• $2 \times 10 \, \text{cm}$ (base)
• $2 \times 11.18 \, \text{cm}$ (triangle height)
• $2 \times 15 \, \text{cm}$ (hypotenuse)
• $3 \times 15 \, \text{cm}$ (rectangular edges)

Final Results:

1. Surface Area: $546.8 \, \text{cm}^2$
2. Edge Lengths:
   • $10 \, \text{cm}$ (2 edges)
   • $11.18 \, \text{cm}$ (2 edges)
   • $15 \, \text{cm}$ (2 edges)
   • $15 \, \text{cm}$ (3 rectangular edges)

Questions for Further Understanding:

1. What is the volume of the triangular prism?
2. How can you generalize the formula for the surface area of a triangular prism?
3. If the prism's dimensions doubled, how would the surface area change?
4. Can you calculate the perimeter of one triangular face?
5. How do you find the lateral surface area alone (excluding triangular faces)?

Tip: Always verify dimensions carefully and use the Pythagorean theorem when dealing with right triangles.