Question

question how many planes are in the figure?

Explanation:

Step1: Identify the base plane

The base plane is \( W \) (containing points \( A, B, C \)).

Step2: Identify the front - back planes

• Front plane: Contains points \( A, B, E, F \).
• Back plane: Contains points \( C, D, E, F \).

Step3: Identify the side planes

• Left - side plane: Contains points \( A, C, D, F \)? No, wait. Let's re - examine. The figure seems to be a prism - like shape with a triangular base? Wait, no, looking at the points: \( A, B, C \) on the base, \( F, D, E \) above. Wait, actually, let's list all possible planes:

1. Plane \( W \) (base: \( A, B, C \))
2. Plane \( ABFE \) (front)
3. Plane \( BCDE \) (right - side? Wait, no, \( B, C, D, E \)? Wait, \( B, C, D, E \): \( B \) to \( C \) to \( D \) to \( E \) to \( B \)? Wait, maybe I made a mistake. Let's look at the triangular - like faces. Wait, the figure has a triangular base? Wait, points \( A, B, C \): maybe triangle \( ABC \), and triangle \( FDE \) above, with rectangles (or parallelograms) connecting them.

Wait, let's correctly identify the planes:

• Plane \( W \) (contains \( A, B, C \))
• Plane \( ABF E \) (contains \( A, B, E, F \))
• Plane \( BCDE \) (contains \( B, C, D, E \))
• Plane \( ACDF \) (contains \( A, C, D, F \))
• Plane \( FDE \) (contains \( F, D, E \))? No, wait, \( F, D, E \) and \( A, B, C \): maybe the figure is a triangular prism? Wait, a triangular prism has 5 planes: 2 triangular bases and 3 rectangular lateral faces. Wait, in the figure, the base plane \( W \) (which is a triangular base? Wait, \( A, B, C \) are on plane \( W \), and \( F, D, E \) are above. So:

1. Plane \( ABC \) (which is plane \( W \))
2. Plane \( FDE \) (the top triangular plane)
3. Plane \( ABFE \) (the front rectangular face)
4. Plane \( BCDE \) (the right - side rectangular face)
5. Plane \( ACDF \) (the left - side rectangular face)

Wait, let's count again. The base is a triangle \( ABC \) on plane \( W \), the top is triangle \( FDE \). Then the three lateral faces: \( ABFE \) (connecting \( AB \) to \( FE \)), \( BCDE \) (connecting \( BC \) to \( DE \)), and \( ACDF \) (connecting \( AC \) to \( FD \)). So in total, we have the base plane \( W \) (triangle \( ABC \)), the top plane \( FDE \), and three lateral planes. Wait, but maybe the base plane \( W \) is considered as one, and then the other four? Wait, no, let's look at the figure again. The points are \( A, B, C \) on the base (plane \( W \)), \( F, D \) above \( C, A \) respectively, and \( E \) in the middle. So:

• Plane 1: \( W \) (contains \( A, B, C \))
• Plane 2: \( ABFE \) (contains \( A, B, E, F \))
• Plane 3: \( BCDE \) (contains \( B, C, D, E \))
• Plane 4: \( ACDF \) (contains \( A, C, D, F \))
• Plane 5: \( FDE \)? No, \( F, D, E \): is \( E \) connected to \( F \) and \( D \)? Yes, from the figure, \( E \) is inside the prism - like shape. Wait, maybe the correct count is 5? Wait, no, let's think of a triangular prism. A triangular prism has 2 triangular bases and 3 rectangular lateral faces, so total 5 planes. Let's confirm with the points:

• Triangular base 1: \( A, B, C \) (plane \( W \))
• Triangular base 2: \( F, D, E \)
• Lateral face 1: \( A, B, F, E \)
• Lateral face 2: \( B, C, D, E \)
• Lateral face 3: \( A, C, D, F \)

Yes, that's 5 planes. Wait, but maybe I miscounted. Wait, the figure shows a prism with a triangular base ( \( A, B, C \)) and a triangular top ( \( F, D, E \)), and three rectangular sides. So the number of planes is 5.

Answer:

5