Question

1. a, b, and i

naming noncoplanar points name all the points that are not
coplanar with the given points.

1. n, k, and l 47. s, p, and m
2. p, q, and n 49. r, s, and l
3. p, q, and r 51. r, k, and n
4. p, s, and k 53. q, k, and l

Explanation:

To solve these problems, we analyze the coplanarity of points in the cube:

Problem 46: N, K, and L

• Points \( N \), \( K \), and \( L \) lie on the top - front - left face (or a face containing these three).
• Points not on this face: \( P \), \( Q \), \( R \), \( S \) (since they are on the bottom or other non - overlapping faces).

Problem 47: S, P, and M

• Points \( S \), \( P \), and \( M \): \( S \) and \( P \) are on the bottom - right - front, \( M \) is on the top - middle. The plane of \( S \), \( P \), \( M \) does not include \( Q \), \( R \), \( N \) (checking the cube's structure: \( Q \) is bottom - left - front, \( R \) is inside/bottom - middle, \( N \) is top - left - front; they don't lie on the same plane as \( S \), \( P \), \( M \)).

Problem 48: P, Q, and N

• Points \( P \), \( Q \), and \( N \): \( P \) (bottom - right - front), \( Q \) (bottom - left - front), \( N \) (top - left - front) lie on a face.
• Points not on this face: \( R \), \( S \), \( L \), \( M \) ( \( R \) is bottom - middle, \( S \) is bottom - right - back, \( L \) is top - right - front, \( M \) is top - middle; they are on different planes).

Problem 49: R, S, and L

• Points \( R \), \( S \), and \( L \): \( R \) (bottom - middle), \( S \) (bottom - right - back), \( L \) (top - right - front) form a plane.
• Points not on this plane: \( P \), \( Q \), \( N \) ( \( P \) is bottom - right - front, \( Q \) is bottom - left - front, \( N \) is top - left - front; they are on a different face).

Problem 50: P, Q, and R

• Points \( P \), \( Q \), and \( R \) lie on the bottom face (or a face containing these three).
• Points not on this face: \( K \), \( L \), \( M \), \( N \) (they are on the top face or related top - side faces).

Problem 51: R, K, and N

• Points \( R \), \( K \), and \( N \): \( R \) (bottom - middle), \( K \) (top - left - front), \( N \) (top - left - front) - wait, re - analyzing: \( R \) is inside the cube (bottom - middle), \( K \) (top - left - front), \( N \) (top - left - front). The plane of \( R \), \( K \), \( N \) does not include \( P \), \( Q \), \( S \), \( L \) ( \( P \) is bottom - right - front, \( Q \) is bottom - left - front, \( S \) is bottom - right - back, \( L \) is top - right - front; they are on different planes).

Problem 52: P, S, and K

• Points \( P \), \( S \), and \( K \): \( P \) (bottom - right - front), \( S \) (bottom - right - back), \( K \) (top - left - front) form a plane.
• Points not on this plane: \( Q \), \( R \), \( M \), \( N \) ( \( Q \) is bottom - left - front, \( R \) is bottom - middle, \( M \) is top - middle, \( N \) is top - left - front; they are on different planes).

Problem 53: Q, K, and L

• Points \( Q \), \( K \), and \( L \): \( Q \) (bottom - left - front), \( K \) (top - left - front), \( L \) (top - right - front) lie on a face.
• Points not on this face: \( P \), \( R \), \( S \), \( M \) ( \( P \) is bottom - right - front, \( R \) is bottom - middle, \( S \) is bottom - right - back, \( M \) is top - middle; they are on different planes).

Final Answers (for each sub - problem):

1. \( P, Q, R, S \)
2. \( Q, R, N \)
3. \( R, S, L, M \)
4. \( P, Q, N \)
5. \( K, L, M, N \)
6. \( P, Q, S, L \)
7. \( Q, R, M, N \)
8. \( P, R, S, M \)

Answer:

To solve these problems, we analyze the coplanarity of points in the cube:

Problem 46: N, K, and L

• Points \( N \), \( K \), and \( L \) lie on the top - front - left face (or a face containing these three).
• Points not on this face: \( P \), \( Q \), \( R \), \( S \) (since they are on the bottom or other non - overlapping faces).

Problem 47: S, P, and M

• Points \( S \), \( P \), and \( M \): \( S \) and \( P \) are on the bottom - right - front, \( M \) is on the top - middle. The plane of \( S \), \( P \), \( M \) does not include \( Q \), \( R \), \( N \) (checking the cube's structure: \( Q \) is bottom - left - front, \( R \) is inside/bottom - middle, \( N \) is top - left - front; they don't lie on the same plane as \( S \), \( P \), \( M \)).

Problem 48: P, Q, and N

• Points \( P \), \( Q \), and \( N \): \( P \) (bottom - right - front), \( Q \) (bottom - left - front), \( N \) (top - left - front) lie on a face.
• Points not on this face: \( R \), \( S \), \( L \), \( M \) ( \( R \) is bottom - middle, \( S \) is bottom - right - back, \( L \) is top - right - front, \( M \) is top - middle; they are on different planes).

Problem 49: R, S, and L

• Points \( R \), \( S \), and \( L \): \( R \) (bottom - middle), \( S \) (bottom - right - back), \( L \) (top - right - front) form a plane.
• Points not on this plane: \( P \), \( Q \), \( N \) ( \( P \) is bottom - right - front, \( Q \) is bottom - left - front, \( N \) is top - left - front; they are on a different face).

Problem 50: P, Q, and R

• Points \( P \), \( Q \), and \( R \) lie on the bottom face (or a face containing these three).
• Points not on this face: \( K \), \( L \), \( M \), \( N \) (they are on the top face or related top - side faces).

Problem 51: R, K, and N

• Points \( R \), \( K \), and \( N \): \( R \) (bottom - middle), \( K \) (top - left - front), \( N \) (top - left - front) - wait, re - analyzing: \( R \) is inside the cube (bottom - middle), \( K \) (top - left - front), \( N \) (top - left - front). The plane of \( R \), \( K \), \( N \) does not include \( P \), \( Q \), \( S \), \( L \) ( \( P \) is bottom - right - front, \( Q \) is bottom - left - front, \( S \) is bottom - right - back, \( L \) is top - right - front; they are on different planes).

Problem 52: P, S, and K

• Points \( P \), \( S \), and \( K \): \( P \) (bottom - right - front), \( S \) (bottom - right - back), \( K \) (top - left - front) form a plane.
• Points not on this plane: \( Q \), \( R \), \( M \), \( N \) ( \( Q \) is bottom - left - front, \( R \) is bottom - middle, \( M \) is top - middle, \( N \) is top - left - front; they are on different planes).

Problem 53: Q, K, and L

• Points \( Q \), \( K \), and \( L \): \( Q \) (bottom - left - front), \( K \) (top - left - front), \( L \) (top - right - front) lie on a face.
• Points not on this face: \( P \), \( R \), \( S \), \( M \) ( \( P \) is bottom - right - front, \( R \) is bottom - middle, \( S \) is bottom - right - back, \( M \) is top - middle; they are on different planes).

Final Answers (for each sub - problem):

1. \( P, Q, R, S \)
2. \( Q, R, N \)
3. \( R, S, L, M \)
4. \( P, Q, N \)
5. \( K, L, M, N \)
6. \( P, Q, S, L \)
7. \( Q, R, M, N \)
8. \( P, R, S, M \)