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The problem is asking for the area of the diagonal cross-section of a rectangular parallelepiped (rectangular prism). Here's how we solve it:

### Given:
- The sides of the base of the rectangular prism are 5 mm and 12 mm.
- The lateral edge (height) is 7 mm.

### Step 1: Diagonal of the Base
The base of the prism is a rectangle. To find the diagonal of this rectangle, we use the Pythagorean theorem:

\[
d_{\text{base}} = \sqrt{(5^2 + 12^2)} = \sqrt{25 + 144} = \sqrt{169} = 13 \, \text{mm}
\]

### Step 2: Diagonal of the Prism
Now, we calculate the diagonal of the entire prism, which acts as the hypotenuse of a right triangle formed by the base diagonal \(d_{\text{base}}\) and the height of the prism (lateral edge). Again, using the Pythagorean theorem:

\[
d_{\text{prism}} = \sqrt{(13^2 + 7^2)} = \sqrt{169 + 49} = \sqrt{218} \approx 14.77 \, \text{mm}
\]

### Step 3: Area of the Diagonal Cross-Section
The diagonal cross-section forms a right triangle where one leg is the lateral edge (7 mm), and the other leg is the diagonal of the base (13 mm). The area \(A\) of this triangle is:

\[
A = \frac{1}{2} \times 13 \times 7 = \frac{1}{2} \times 91 = 45.5 \, \text{mm}^2
\]

Thus, the area of the diagonal cross-section is **45.5 mm²**.

Let me know if you want more details or explanations!

### 5 Related Questions:
1. How is the diagonal of a prism's base calculated for non-rectangular bases?
2. How does the Pythagorean theorem apply to three-dimensional shapes?
3. What are other methods to calculate the diagonal cross-section area of complex prisms?
4. How does the area change if the dimensions of the base are doubled?
5. How would you calculate the volume of the same rectangular prism?

**Tip:** In three-dimensional geometry, always look for ways to break down problems into smaller two-dimensional problems using triangles and the Pythagorean theorem.

The sides of the base of a rectangular parallelepiped are 5 mm and 12 mm, and the lateral edge is 7 mm. Find the area of the diagonal cross-section.

Calculate the area of the diagonal cross-section of a rectangular parallelepiped with sides 5 mm and 12 mm, and a lateral edge of 7 mm using the Pythagorean theorem. Learn how to apply geometry concepts for three-dimensional figures.

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