planes a and b both intersect plane s. which statements are true based on the diagram? select three options. ☐ points n and k are on plane a and plane s. ☐ points p and m are on plane b and plane s. ☐ point p is the intersection of line n and line g. ☐ points m, p, and q are noncollinear. ☐ line d intersects plane a at point n.

Question

Accepted Answer

To solve this, we analyze each statement using the diagram:

### Statement 1: Points N and K are on plane A and plane S.
- Plane A and plane S intersect. Points N and K lie on the intersection region (the overlapping area of plane A and plane S). So, they are on both planes. This statement is true.

### Statement 2: Points P and M are on plane B and plane S.
- Plane B (the vertical plane) and plane S (the lower horizontal plane) intersect? Wait, looking at the diagram, plane B (with line g and point P) and plane S (lower horizontal) – point M is on plane S (lower horizontal) and line n, but point P: is P on plane B? Plane B is the vertical plane (with line g). Wait, maybe misinterpretation. Wait, plane B: the vertical plane (the one with line g and point P? Wait, no, the lower horizontal is plane S, and the vertical plane (let's see: plane A is upper horizontal, plane S (upper) and plane S (lower)? Wait, maybe the diagram has two horizontal planes (A and lower S) and a vertical plane (B) intersecting them. Wait, re-examining: Plane A (upper horizontal), plane S (upper, the blue part), and lower plane S? Wait, maybe the diagram has plane A (upper horizontal), plane B (vertical), and two horizontal planes (upper S and lower S). Wait, no, the labels: "Planes A and B both intersect plane S". So plane S is one plane? Wait, maybe the diagram has plane A (horizontal, upper), plane B (vertical), and plane S (maybe the lower horizontal? Or the vertical? No, the text says "Planes A and B both intersect plane S". So plane S is a single plane, intersected by A and B.

Wait, maybe the diagram: Plane A is the upper horizontal plane, plane B is the vertical plane (with line g and point P), and plane S is the lower horizontal plane? No, the points: N and K are on plane A (upper horizontal) and plane S (the vertical? No, the blue area is plane S? Wait, the diagram shows: upper horizontal plane A, vertical plane S (blue), and lower horizontal plane (labeled s? Maybe typo, plane S). Then plane B is another vertical plane? Wait, the problem says "Planes A and B both intersect plane S". So plane S is a plane, intersected by A and B.

Let's re-express:

- Plane A: upper horizontal.
- Plane B: vertical (with line g and point P).
- Plane S: maybe the vertical plane (blue) or the lower horizontal. Wait, the points:

- Point N and K: on plane A (upper horizontal) and plane S (the vertical blue plane, since they are in the overlapping region). So yes, on both A and S. So statement 1 is true.

- Statement 2: Points P and M are on plane B and plane S. Plane B: vertical (with line g). Plane S: lower horizontal? Point M is on plane S (lower horizontal) and line n. Point P: is P on plane B? If plane B is the vertical plane with line g, then P is on line g (so on plane B) and on plane S (lower horizontal). Point M: is M on plane B? No, M is on line n (on plane S, lower horizontal) and not on plane B. So statement 2 is false.

### Statement 3: Point P is the intersection of line n and line g.
- Line n is horizontal (on lower plane S), line g is vertical (on plane B). They intersect at point P. So this is true.

### Statement 4: Points M, P, and Q are noncollinear.
- Collinear points lie on a straight line. Points M, P, Q: M---P---Q on line n (horizontal line on lower plane S). Wait, line n passes through M, P, Q? If line n is a straight line, then M, P, Q are collinear. Wait, but the option says "noncollinear" (not collinear). Wait, maybe I missee. Wait, the diagram: line n (with arrow) has M, P, Q. So they are on a straight line, so collinear. But the option says "noncollinear" – that would be false? Wait, no, maybe the diagram shows M, P, Q not on a straight line? Wait, the user's diagram: "n" is a line with M, P, Q. If it's a straight line, they are collinear. But maybe the diagram has them as noncollinear? Wait, no, the option says "Points M, P, and Q are noncollinear" – if they are on a straight line, that's false. But maybe I made a mistake. Wait, let's check again.

Wait, the problem says "Select three options". Let's check other statements.

### Statement 5: Line d intersects plane A at point N.
- Line d (with arrow) is vertical, intersecting plane A (upper horizontal) at point N. So this is true.

Wait, let's re-express:

1. Points N and K are on plane A and plane S: True (they are in the intersection of A and S).
2. Points P and M are on plane B and plane S: False (M is not on plane B).
3. Point P is the intersection of line n and line g: True (line n and line g meet at P).
4. Points M, P, and Q are noncollinear: Wait, if M, P, Q are on line n, they are collinear, so this is False. But the problem says to select three options. Wait, maybe I misinterpret plane B.

Wait, maybe plane B is the lower vertical? No, let's re-express the diagram:

- Plane A: upper horizontal (with points L, N, K).
- Plane S: vertical (blue, intersecting A).
- Plane B: lower vertical (with line g and point P), intersecting lower plane S (horizontal).

Wait, maybe "plane S" is the vertical plane (blue) and another plane? No, the problem says "Planes A and B both intersect plane S" – so plane S is one plane, intersected by A and B.

Alternative approach:

- Statement 1: True (N and K are on A and S).
- Statement 3: True (P is intersection of n and g).
- Statement 5: True (line d intersects A at N).

Wait, but the problem says "Select three options". Let's check:

1. Points N and K are on plane A and plane S: True.
3. Point P is the intersection of line n and line g: True.
5. Line d intersects plane A at point N: True.

Wait, but what about statement 4? If M, P, Q are on line n, they are collinear, so statement 4 is False. Statement 2: Points P and M are on plane B and plane S. Plane B: if plane B is the vertical plane (with line g), then P is on B (line g) and S (lower plane), but M is on S (lower plane) and line n, not on B. So statement 2 is False.

So the three true statements are:

- Points N and K are on plane A and plane S.
- Point P is the intersection of line n and line g.
- Line d intersects plane A at point N.

Wait, but let's confirm:

- Line d: the vertical line, intersects plane A (upper horizontal) at N. Correct.
- Point P: line n (horizontal) and line g (vertical) intersect at P. Correct.
- Points N and K: on plane A (upper horizontal) and plane S (vertical blue plane). Correct.

So the three true options are:

1. Points N and K are on plane A and plane S.
3. Point P is the intersection of line n and line g.
5. Line d intersects plane A at point N.

Wait, but the problem's options:

1. Points N and K are on plane A and plane S. – True.
2. Points P and M are on plane B and plane S. – False (M not on B).
3. Point P is the intersection of line n and line g. – True.
4. Points M, P, and Q are noncollinear. – False (they are collinear on line n).
5. Line d intersects plane A at point N. – True.

So the three true ones are 1, 3, 5.

# Brief Explanations:
- **Points N and K are on plane A and plane S**: N and K lie in the intersection of plane A (upper horizontal) and plane S (vertical), so they belong to both.
- **Point P is the intersection of line n and line g**: Line $ n $ (horizontal) and line $ g $ (vertical) meet at $ P $, so $ P $ is their intersection.
- **Line d intersects plane A at point N**: Line $ d $ (vertical) crosses plane A (upper horizontal) at point $ N $.

# Answer:
- Points N and K are on plane A and plane S.
- Point P is the intersection of line n and line g.
- Line d intersects plane A at point N.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Statement 1: Points N and K are on plane A and plane S.

Statement 2: Points P and M are on plane B and plane S.

Statement 3: Point P is the intersection of line n and line g.

Statement 4: Points M, P, and Q are noncollinear.

Answer: