which planes contain point w? select all that apply. a) plane b b) plane vwy c) plane vwx d) plane m e) plane ryv f) plane vwz

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# Brief Explanations:
- **Plane B**: Point W is on the blue plane labeled B, so plane B contains W.
- **Plane VWY**: Points V, W, Y are colinear (on line n), so they define a plane that contains W.
- **Plane VWX**: Points V, W, X: W is on line n (with V, Y), X is another point. But from the diagram, W, V, Y are on a vertical line, and X is on a horizontal line with Y, R. Wait, no—actually, plane VWX: Let's check. Wait, maybe I misread. Wait, plane VWY: V, W, Y are on the same line (n), so the plane VWY would include that line. Also, plane B is the large blue plane, which contains W. Let's re-examine each option:
  - **A (Plane B)**: The blue plane includes W, so yes.
  - **B (Plane VWY)**: V, W, Y are on line n, so the plane through these three (collinear) includes W.
  - **C (Plane VWX)**: X, Y are on a horizontal line, W is on vertical line from Y. So X, W, V: Do they lie on a plane? Wait, X is on the horizontal line with Y, W is below Y on line n, V is below W. So X, W, V: Let's see, X is (let's say) left of Y, W is below Y, V below W. So three non - collinear points? Wait, no—X, Y, W: X and Y are on a horizontal line, Y and W on vertical. So X, Y, W are non - collinear, but plane VWX: V, W, X. V and W are on line n, X is on line with Y. So does X, V, W lie on a plane? Maybe, but let's check the diagram again. Wait, the blue plane is B, which contains all the points on the parallelogram. So plane B (the blue one) contains W, X, Y, R, V, etc. Then plane VWY: V, W, Y are on line n, so the plane VWY is part of plane B? Wait, no—any three collinear points lie on infinitely many planes, but also on the plane B. Wait, maybe I made a mistake. Let's go back:
    - **Plane B**: The large blue parallelogram. Point W is inside (or on) it, so yes.
    - **Plane VWY**: Points V, W, Y are on the same vertical line (n), so the plane defined by these three (collinear) points includes W.
    - **Plane VWX**: X, W, V. X is on the horizontal line with Y, W is below Y, V below W. So X, W, V: Are they coplanar? Since X, Y, W are coplanar (in plane B), and V is on line WY, so X, W, V are also coplanar (in plane B). Wait, but maybe the question is about the named planes. Wait, the options:
      - A: Plane B (the big blue one) – contains W.
      - B: Plane VWY – V, W, Y are on a line, so the plane through them (which is part of plane B) contains W.
      - C: Plane VWX – X, W, V: X is on the horizontal line, W and V on vertical. So X, W, V are coplanar (in plane B), so this plane contains W? Wait, maybe I was wrong earlier. Wait, let's think about the definition of a plane: three non - collinear points define a plane, but a line and a point not on the line define a plane. For plane VWX: V and W are on line n, X is not on line n, so X, V, W define a plane, which is the same as plane B (since X, Y, W are in plane B, and V is on line WY). So plane VWX is part of plane B, so it contains W. Wait, but maybe the diagram shows that plane VWX is a different plane? No, the blue plane is B, which contains all these points. Wait, maybe the correct options are A, B, C? Wait, no, let's check again.
      - Plane m: Plane m is the horizontal line through Y and R. W is below Y, so not on plane m.
      - Plane RYV: R is on line m, Y is on line m and n, V is on line n. So R, Y, V: R is on m, Y is the intersection of m and n, V is on n. So R, Y, V are coplanar (in plane B), but does W lie on plane RYV? W is on line n between Y and V, so yes! Wait, R, Y, V: Y is on m and n, R is on m, V is on n. So the plane RYV contains line m (RY) and line n (YV), so it's the same as plane B. So W is on line YV, so W is on plane RYV. Wait, I missed that earlier.
      - Plane VWZ: Z is below R, on line l. W is on line n (vertical from Y), Z is on line l (vertical from R). So W and Z are on different vertical lines, so plane VWZ would not contain W? Wait, no—V, W, Z: V and W are on line n, Z is on line l. So three points: V, W (collinear), Z (not on line n). So the plane VWZ would contain line n and point Z, but W is on line n, so yes, W is on plane VWZ? Wait, no, because line n and line l are skew? No, in the diagram, line n (through Y, W, V) and line l (through R, Z) are parallel? Wait, the diagram shows line n (with Y, W, V) and line l (with R, Z) as both vertical (assuming the arrows are vertical), and line m (through Y, R) as horizontal. So line n and line l are parallel, so they don't intersect. So plane VWZ: V, W (on line n), Z (on line l). Since line n and line l are parallel, the plane containing line n and point Z is a plane that contains W (since W is on line n). Wait, but maybe the diagram is a parallelogram, so line n and line l are parallel, and plane B is the parallelogram, so plane B contains both lines n and l? Wait, no—if the parallelogram has sides m (horizontal) and n (vertical), and the other side parallel to n would be l? Wait, maybe I misinterpret the diagram. Let's start over:

The blue region is plane B. Point W is inside plane B (on the vertical line n through Y, with V below and X left of Y on line m).

- **Option A (Plane B)**: Yes, W is in plane B.
- **Option B (Plane VWY)**: V, W, Y are colinear (on line n), so the plane through them (which is part of plane B) contains W.
- **Option C (Plane VWX)**: X, W, V. X is on line m (horizontal), W and V on line n (vertical). So X, W, V are coplanar (in plane B), so this plane contains W.
- **Option D (Plane m)**: Plane m is the horizontal line through Y and R. W is below Y, so not on plane m.
- **Option E (Plane RYV)**: R is on line m, Y is on m and n, V is on n. So R, Y, V: Y is the intersection of m and n, R is on m, V is on n. So the plane RYV contains line m (RY) and line n (YV), so it's plane B, and W is on line YV, so W is on plane RYV.
- **Option F (Plane VWZ)**: Z is on line l (vertical through R), W is on line n (vertical through Y). Lines n and l are parallel (since the figure is a parallelogram), so the plane containing V, W (on n) and Z (on l) is a plane that contains line n and point Z. Since W is on line n, W is on this plane? Wait, but in the diagram, is Z in plane B? Z is below R, which is on plane B, so Z is not in plane B. So plane VWZ would be a different plane, not containing W? Wait, no—if we consider the 3D space, but the diagram is a plane (B) with lines perpendicular to it? Wait, maybe the lines l and n are perpendicular to plane B? No, the arrows for l and n are inside the plane B (the blue parallelogram), so l and n are in plane B. Wait, R is on plane B, Z is below R, so Z is not in plane B. So W is in plane B, Z is not, so plane VWZ does not contain W (because Z is not in plane B, and W is in plane B, so the plane through V, W, Z would not be plane B, and W is in plane B, Z is not, so W is not in plane VWZ).

Wait, I think I made a mistake with option E. Let's re - examine plane RYV: R, Y, V. R is on line m, Y is on m and n, V is on n. So the plane RYV is the same as plane B, because it contains two intersecting lines (m and n) which define plane B. So W is on line n (between Y and V), so W is on plane RYV.

And option C: Plane VWX. X, W, V. X is on line m, W and V on line n. So X, W, V: X and Y are on line m, Y and W on line n, so X, Y, W are non - collinear, defining plane B, and V is on line YW, so X, W, V are also in plane B, so plane VWX is part of plane B, containing W.

So the correct options are A, B, C, E? Wait, no, let's check the diagram again. The blue plane is B. Point W is on the vertical line n (Y - W - V), horizontal line m (X - Y - R). So:

- Plane B: Contains W (yes).
- Plane VWY: V, W, Y are on line n, so plane VWY (which is a subset of plane B) contains W (yes).
- Plane VWX: X, W, V. X is on m, W on n, V on n. So X, W, V: X and Y are on m, Y and W on n, so X, Y, W are coplanar (plane B), and V is on YW, so X, W, V are coplanar (plane B) (yes).
- Plane RYV: R, Y, V. R on m, Y on m and n, V on n. So R, Y, V: m and n intersect at Y, so plane RYV is plane B, and W is on YV, so W is on plane RYV (yes).
- Plane m: Is plane m the line or the plane? Wait, the option says "plane m"—if m is a line, a plane named after a line is not standard, but maybe in the diagram, plane m is the horizontal plane? No, the diagram shows m as a line. Wait, maybe the options have some errors, but based on the diagram:

Wait, the original diagram: the blue region is plane B. The lines: m is horizontal (Y - R), n is vertical (Y - W - V), l is vertical (R - Z).

So:

- **A (Plane B)**: Yes, W is in plane B.
- **B (Plane VWY)**: V, W, Y are colinear, so plane VWY contains W (yes).
- **C (Plane VWX)**: X, W, V: X is on m, W on n, V on n. So X, W, V are coplanar (in plane B) (yes).
- **D (Plane m)**: Plane m—if m is a line, a plane can't be a line. Maybe it's a typo, but W is not on line m (W is below Y, line m is through Y and R), so no.
- **E (Plane RYV)**: R, Y, V: R on m, Y on m and n, V on n. So R, Y, V are coplanar (plane B), and W is on YV, so W is on plane RYV (yes).
- **F (Plane VWZ)**: Z is below R, not in plane B, W is in plane B, so W is not in plane VWZ (no).

But maybe the intended answers are A, B, C, E? Wait, but let's check the standard definition: a plane is defined by three non - collinear points or a line and a point not on the line.

For plane VWY: collinear points, so it's a valid plane (infinite planes, but in the context, it's part of plane B).

For plane VWX: X, W, V—X is not on line VW, so they define a plane (plane B).

For plane RYV: R, Y, V—R is not on line YV, so they define plane B.

So the correct options are A, B, C, E? Wait, but maybe the diagram is simpler. Let's look at the points:

- W is on the blue plane (B), so A is correct.
- W is on line VWY (V - W - Y), so B is correct.
- W is on line VW, and X is a point such that X, V, W are coplanar (since X is on the same plane B), so C is correct.
- Plane m: line m, W is not on line m, so D is incorrect.
- Plane RYV: R, Y, V—W is on YV, so W is on this plane, so E is correct.
- Plane VWZ: Z is not on plane B, W is on plane B, so F is incorrect.

But maybe the answer is A, B, C, E. Wait, but let's check with the initial diagram again. The blue plane is B, so A is correct. Plane VWY: V, W, Y are on a vertical line, so the plane through them (which is the same as the plane containing that line and any other point, but in this case, it's part of plane B) contains W. Plane VWX: X, W, V—X is on the horizontal line, W and V on vertical, so they are coplanar. Plane RYV: R, Y, V—RY is horizontal, YV is vertical, so the plane through them is plane B, and W is on YV, so W is on it.

So the correct options are A, B, C, E. Wait, but maybe I made a mistake with E. Let's see: RYV—R, Y, V. Y is the intersection of RY (line m) and YV (line n). So plane RYV is the plane containing lines m and n, which is plane B. So W is on line n (YV), so W is on plane RYV. Yes.

And plane VWX: X, W, V. X is on line m, W on line n, V on line n. So X, W, V: X and Y are on line m, Y and W on line n, so X, Y, W are non - collinear, defining plane B, and V is on YW, so X, W, V are also in plane B, so plane VWX contains W.

So the correct options are A, B, C, E.
# Answer:
A) plane B, B) plane VWY, C) plane VWX, E) plane RYV

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