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We are given a square-based pyramid \( P-ABCD \), where the base \( ABCD \) is a square with side length 3. The point \( P \) lies directly above the center of the square, and the height \( PD \) is perpendicular to the plane of \( ABCD \).

The problem consists of two parts:

### (1) Prove: \( BC \perp \text{Plane } CDP \)
To solve this, we need to establish that line \( BC \) is perpendicular to the plane formed by points \( C, D, \) and \( P \). We can utilize vector geometry or spatial reasoning for this proof, considering the perpendicularity of vectors.

### (2) If the angle between line \( AD \) and \( BP \) is \( 60^\circ \), find the length of \( DP \).

We are given that the angle between lines \( AD \) and \( BP \) is \( 60^\circ \). Using this information, we can apply geometric relationships, potentially utilizing the cosine rule or vector dot product, to find the length of \( DP \).

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Let's proceed with part (1).

Would you like a detailed step-by-step solution for both parts? Additionally, here are 5 related questions for expanding understanding:

1. How can we use vectors to describe perpendicularity in 3D geometry?
2. What is the importance of the height \( PD \) in establishing perpendicularity?
3. How does the angle between two lines help in determining unknown lengths in geometric problems?
4. Could trigonometry or the Pythagorean theorem be applied here?
5. What other methods could be used to prove perpendicularity in three-dimensional space?

**Tip**: When dealing with perpendicular planes or lines, drawing auxiliary projections or using vector operations helps clarify the relationships.

In the square-based pyramid P-ABCD, where ABCD is a square with side length 3, and PD is perpendicular to the plane ABCD:
1. Prove that BC is perpendicular to the plane CDP.
2. If the angle between AD and BP is 60°, find the length of DP.

Solve this advanced geometry problem involving a square-based pyramid. Learn how to prove perpendicularity between line BC and the plane CDP, and calculate the length of DP using trigonometry, given an angle of 60° between lines AD and BP. Suitable for Grades 10-12.

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