https://file.aiquickdraw.com/mathbot/file/1729147892512wcftugvw.jpg

The problem in the image is asking to find the lateral surface area of a right prism, where the base is a right triangle with legs measuring 3 cm and 4 cm, and the height of the prism is 4 cm. Let's break down the steps to solve it.

### Steps:

1. **Determine the Perimeter of the Base:**
   The base is a right triangle with legs 3 cm and 4 cm. The hypotenuse can be found using the Pythagorean theorem:

   \[
   c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{cm}
   \]

   The perimeter of the triangular base is the sum of all its sides:

   \[
   P = 3 + 4 + 5 = 12 \, \text{cm}
   \]

2. **Calculate the Lateral Surface Area:**
   The lateral surface area of a prism is given by the formula:

   \[
   A_{\text{lateral}} = P \times h
   \]

   where \(P\) is the perimeter of the base and \(h\) is the height of the prism. Given that the height of the prism is 4 cm:

   \[
   A_{\text{lateral}} = 12 \times 4 = 48 \, \text{cm}^2
   \]

### Final Answer:
The lateral surface area of the prism is \( \boxed{48 \, \text{cm}^2} \), which corresponds to option **Д**.

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Would you like a more detailed explanation, or do you have any questions?

Here are five related questions to expand on this concept:

1. How would the lateral surface area change if the height of the prism were doubled?
2. What would be the total surface area of the prism, including the bases?
3. How do you find the volume of this prism?
4. What is the lateral surface area of a cylinder with the same height and perimeter as this prism?
5. How would the lateral surface area change if the triangle's legs were halved?

**Tip**: In right prisms, the lateral surface area is always found by multiplying the perimeter of the base by the height of the prism.

Вічне ребро прямої призми, в основі якої лежить прямокутний трикутник, дорівнює 4 см. Знайдіть площу бічної поверхні призми, якщо катети трикутника дорівнюють 3 см і 4 см.

Learn how to find the lateral surface area of a right prism with a right triangle base where the legs are 3 cm and 4 cm, and the prism's height is 4 cm. Use the Pythagorean theorem to calculate the hypotenuse and determine the perimeter of the base. Suitable for students in Grades 8-10.

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