Question

for questions 1-2, use figure 1.

1. which planes contain point w? select all that apply.

a) plane b
b) plane vwy
c) plane vwx
d) plane m
e) plane rvv
f) plane vwz

1. which three points are collinear? select all that apply.

a) points w, x, and r
b) points x, r, and z
c) points w, v and y
d) points x, y, and r

Explanation:

Question 1

• Plane B: The diagram shows point W is on the plane labeled B, so Plane B contains W.
• Plane VWY: Point W is on the line with V and Y, so this plane (defined by V, W, Y) contains W.
• Plane VWX: Point W is on the line with V and also on the line with X (since X, Y are on a horizontal line and W is on the vertical line through Y), so this plane (V, W, X) contains W.
• Plane m: Plane m is the horizontal plane (line m is on it) and W is on the same flat surface (plane B which includes plane m? Wait, looking at the diagram, plane m is part of the plane B? Wait, no, plane m is a horizontal plane (line m) and point W is on the vertical line n through Y. Wait, maybe I misread. Wait, the diagram: plane B is the big plane, line m is horizontal on plane B, line n is vertical on plane B (through Y, W, V), line ℓ is perpendicular to plane B at R. So:
• Plane B: contains all points on the flat surface, so W is on B.
• Plane VWY: V, W, Y are colinear (on line n), so the plane through them (which is part of plane B) contains W.
• Plane VWX: V, W, X – X is on line XY (horizontal), W is on line YV (vertical), so X, Y, W, V are coplanar (plane B), so plane VWX (X, W, V) is part of plane B, contains W.
• Plane m: Plane m is the plane with line m (through R, Y, X? Wait, line m is horizontal, through R and Y? Wait, the diagram: X←Y→R (line m?), and Y↓W↓V (line n). So plane m: if line m is the horizontal line through Y and R, then plane m would be the plane containing line m. But W is on line n (vertical), so is W on plane m? No, because line n is perpendicular to line m (since one is horizontal, one vertical on plane B). Wait, maybe plane m is the same as plane B? No, maybe plane m is a sub - plane. Wait, maybe I made a mistake. Wait, the options:
• Plane B: yes, W is on plane B.
• Plane VWY: V, W, Y are on a line, so the plane through them (which is plane B) contains W.
• Plane VWX: X, W, V – X is on line XY (horizontal), W on YV (vertical), so they are coplanar (plane B), so plane VWX contains W.
• Plane m: If plane m is the plane with line m (through Y, R), then W is not on plane m (since W is below Y on line n), so D is out.
• Plane RYV: R, Y, V – R is on line m, Y on line m and n, V on line n. So R, Y, V: R to Y is horizontal, Y to V is vertical, so they are coplanar (plane B), but does W lie on plane RYV? Plane RYV: points R, Y, V. W is on YV, so yes! Wait, YV is a line, so any plane containing R, Y, V will contain the line YV, so W (on YV) is in plane RYV. Wait, I missed that.
• Plane VWZ: Z is below ℓ, outside plane B, so plane VWZ would not contain W (since W is on plane B, Z is not). So F is out.
• Wait, let's re - evaluate:
• Plane B: Yes, W is on the flat plane B.
• Plane VWY: V, W, Y are colinear, so the plane through them (which is plane B) contains W.
• Plane VWX: X, W, V – X is on the horizontal line, W on vertical, so they are coplanar (plane B), so plane VWX contains W.
• Plane RYV: R, Y, V – YV is a line, W is on YV, and R is on the horizontal line. So the plane through R, Y, V will contain the line YV, so W is in this plane.
• Wait, maybe my initial analysis was wrong. Let's look at the diagram again:
• Line n: Y, W, V (vertical)
• Line m: X, Y, R (horizontal)
• Plane B: the big plane containing lines m and n.
• Plane VWY: V, W, Y (line n), so plane is the plane containing line n (which is part of plane B), so contains W.
• Plane VWX: V, W, X – X is on line m, W on line n, V on line n. So X, W, V: X is (x,y,0), W…

• Option A: Points W, X, and R – W is on line n (vertical), X and R are on line m (horizontal). These three points are not colinear (line n and line m intersect at Y, not passing through W, X, R in a straight line).
• Option B: Points X, R, and Z – X and R are on line m (horizontal), Z is on line ℓ (perpendicular to plane B). These three points are not colinear.
• Option C: Points W, V and Y – W, V, and Y are on the same vertical line n (as seen in the diagram: Y is above W, W above V), so they are colinear.
• Option D: Points X, Y and R – X, Y, and R are on the same horizontal line m (X←Y→R), so they are colinear.

Answer:

A. Plane B, B. Plane VWY, C. Plane VWX, E. Plane RYV

Question 2