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We are given a diagram of a triangular pyramid where the base lies on a plane denoted as \(\alpha\). There are two slopes, \(AD\) and \(DC\), forming angles with the plane. Specifically:

- \(AD\) forms an angle of \(30^\circ\) with the plane \(\alpha\),
- \(DC\) forms an angle of \(45^\circ\) with the plane \(\alpha\),
- The perpendicular from point \(D\) to the plane \(\alpha\) is \(DB = 35\) cm.

We need to calculate the lengths of the slopes \(AD\) and \(DC\).

### Solution Approach:
We can apply trigonometric relations, specifically using the cosine of the angles to calculate the lengths of the slopes.

1. **For \(AD\):**

   The relationship between the height \(DB\) and the slope \(AD\) is given by:
   \[
   \cos(30^\circ) = \frac{DB}{AD}
   \]
   Solving for \(AD\):
   \[
   AD = \frac{DB}{\cos(30^\circ)} = \frac{35}{\cos(30^\circ)}
   \]
   Since \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), we get:
   \[
   AD = \frac{35}{\frac{\sqrt{3}}{2}} = \frac{35 \times 2}{\sqrt{3}} = \frac{70}{\sqrt{3}} \approx 40.41 \, \text{cm}
   \]

2. **For \(DC\):**

   Similarly, for \(DC\), we use the relation:
   \[
   \cos(45^\circ) = \frac{DB}{DC}
   \]
   Solving for \(DC\):
   \[
   DC = \frac{DB}{\cos(45^\circ)} = \frac{35}{\cos(45^\circ)}
   \]
   Since \(\cos(45^\circ) = \frac{1}{\sqrt{2}}\), we get:
   \[
   DC = \frac{35}{\frac{1}{\sqrt{2}}} = 35 \times \sqrt{2} \approx 49.50 \, \text{cm}
   \]

### Final Answer:
- \(AD \approx 40.41\) cm
- \(DC \approx 49.50\) cm

Would you like further details or explanations? Here are some follow-up questions:

1. What trigonometric functions are used for finding lengths in triangles?
2. How does the angle between a line and a plane affect the line's projection?
3. What is the relationship between the angle and the length of the hypotenuse in right triangles?
4. How do you apply cosine values of standard angles?
5. How does a perpendicular relate to calculating the lengths of oblique lines?

**Tip:** For right-angle triangles, the cosine of an angle helps to find the ratio between the adjacent side and the hypotenuse.

Похила AD з площиною α утворює кут 30°, а похила DC з площиною α утворює кут 45°. Довжина перпендикуляра DB дорівнює 35 см. Обчисли довжини обох похилих.

Learn how to calculate the lengths of slopes AD and DC using trigonometry. Given that slopes form angles of 30° and 45° with a plane and a perpendicular length of 35 cm, this problem applies cosine functions to determine the lengths. Suitable for high school students in Grades 9-12.

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