https://file.aiquickdraw.com/mathbot/file/1727730825951oawtlqbn.jpg

The diagram represents a pyramid with base \(ABCD\) and apex \(P\), intersected by a plane passing through points \(K\), \(L\), and \(M\), dividing the figure into two sections. The task is in two parts:

1. **Part (a):** 
   Determine if the planes \((KLM)\) and \((PQT)\) are parallel or intersecting.

2. **Part (b):** 
   Construct the intersection of the base plane with the cutting plane \((KLM)\), which refers to finding the trace of the section in the base plane.

### Solution:

1. **Part (a):** 
   To find if planes \((KLM)\) and \((PQT)\) are parallel or intersecting, we need to analyze their points. If three points \(K\), \(L\), and \(M\) lie on a plane, and points \(P\), \(Q\), and \(T\) lie on another plane, the relationship between the planes can be deduced based on their normal vectors or through common points or lines.

   From the diagram, there is no direct evidence of the planes being parallel, and it seems they intersect. Hence, the planes \((KLM)\) and \((PQT)\) **intersect** in space.

2. **Part (b):** 
   To construct the trace of the section plane \((KLM)\) in the base plane \(ABCD\), we extend lines \(KM\) and \(LM\) to meet the edges of the base \(ABCD\). These intersections will define the trace in the base.

   Here’s the process:
   - Extend line \(KM\) until it intersects the line segment \(AB\).
   - Extend line \(LM\) until it intersects line segment \(CD\).
   - The resulting line in the base plane \(ABCD\) will be the trace of the cutting plane \((KLM)\).

Would you like more details on the construction of these intersections or other parts of the solution?

### Follow-up Questions:
1. How do you determine if two planes are parallel in 3D space?
2. Can you explain how the trace of a plane in a base polygon is constructed geometrically?
3. What are the properties of the intersection of two planes?
4. How do you find the equation of a plane from three points?
5. How can the distance between two planes be calculated if they are parallel?

### Tip:
When constructing the intersection of a plane with a base, always extend the lines from the points of the plane to the edges of the base for accurate geometric intersection points.

a) Determine if planes (KLM) and (PQT) are parallel or intersecting. b) Construct the intersection of the base plane with the cutting plane (KLM).

Solve a geometry problem involving the intersection of planes KLM and PQT. Learn how to determine if two planes are parallel or intersecting and how to construct the intersection of a cutting plane with a base polygon. This problem covers key 3D geometry concepts suitable for high school students.

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