Space

In the world of video game design, objects are defined by their dimensions. State how many dimensions ($0$, $1$, $2$ or $3$) the following items that we find in a room have:

1. a) The cable of a pair of headphones stretched out.

1. b) The flat screen of a monitor (not counting its thickness).

1. c) A box where you keep your console games.

1. d) A point of light on the ceiling.

Imagine that the walls, floor, and ceiling of your classroom are planes. State the relative position of:

1. a) The floor plane and the ceiling plane.
2. b) The plane of the whiteboard wall and the plane of the window wall (which is next to it).
3. c) A line formed by the corner between two walls and the floor plane.

Look at the edges of this cube and determine the relative position (parallel, intersecting, or skew) of the following pairs of lines containing the edges:

1. a) The line passing through points $A$ and $B$ with respect to the line passing through $C$ and $D$.
2. b) The line passing through $A$ and $B$ with respect to the line passing through $B$ and $C$.
3. c) The line passing through $A$ and $B$ with respect to the line passing through $G$ and $H$.

In space, there are lines that are not parallel but never intersect. These lines are called skew lines. Look at the drawing of the skate ramp and state their relative positions:

1. a) The top edge of the ramp ($r$) and the base line on the ground ($s$).
2. b) The vertical edge of the handrail ($t$) and line $s$.

Consider a cube or regular hexahedron with bottom base $ABCD$ and top base $EFGH$, where the vertical edges are $AE$, $BF$, $CG$, and $DH$. Identify the relative positions of the indicated edges:

1. a) Edge $AB$ relative to edge $EF$.
2. b) Edge $AD$ relative to edge $CG$.
3. c) Edge $GH$ relative to edge $AB$.

Draw in your exercise book (or describe) the position of a line and a plane in the following cases:

1. a) The line is parallel to the plane (think of a fluorescent light on the ceiling and the floor).
2. b) The line is contained in the plane (think of a line painted on the floor).
3. c) The line intersects the plane (think of a pencil going through a sheet of paper).

You can do it on paper and send a photo with your answer.

In a room with perpendicular walls, consider one of the top corners of the ceiling as the origin of points. If we draw an imaginary line across the room from that corner to the opposite corner of the floor (diagonal of the room):

1. a) What relative position does this line have with respect to the floor plane?
2. b) Does this line intersect the line formed by the junction of two side walls? Justify your answer.
3. c) How many points in common does the diagonal have with the plane of one of the walls?

In designing an advertising marquee shaped like a right triangular prism, we want to analyse the relationship between its planes. If we consider plane $\pi$ containing the lower base of the prism:

1. a) State which faces of the prism generate planes that intersect plane $\pi$.
2. b) State which edges of the prism are parallel to plane $\pi$.
3. c) What 2D shape is formed at the intersection of a lateral face with the plane of the base?

Imagine you are designing a structure for an adventure park. You have a giant cube whose vertices are labelled as $A, B, C, D$ (lower base) and $E, F, G, H$ (upper base, such that $E$ is above $A$, $F$ above $B$, etc.).

If we draw a straight line passing through the midpoints of the edges $AE$ and $CG$, and another straight line connecting the vertices $B$ and $H$:

1. a) State the relative position of these two lines in space. Justify whether they intersect, are parallel, or are skew.
2. b) If we draw a plane passing through points $B$, $D$, and $H$, what type of cross-section (2D shape) is formed when cutting the cube?
3. c) How many edges of the cube intersect the plane defined by face $ADHE$?

Many geometric solids are generated by rotating a plane figure around an axis. These are called solids of revolution. In the drawing you see three objects with their axis of rotation. Identify which figure (rectangle, right triangle or semicircle) generated each of them when rotated around that axis:

1. a) A cylindrical soda can.
2. b) A traffic cone.
3. c) A pétanque ball.

Geometric solids whose faces are all polygons are called polyhedra. In any simple polyhedron, a famous relationship called Euler's formula holds: $Faces + Vertices = Edges + 2$.

Check whether this formula holds for a cube, knowing that it has $6$ faces, $8$ vertices, and $12$ edges. Write down the mathematical calculation.

A cut is made through a right cylinder of radius $r = 4$ cm and height $h = 10$ cm using a plane. Describe what 2D shape is obtained if:

1. a) The plane is parallel to the bases of the cylinder.
2. b) The plane is perpendicular to the bases and passes through their centres.
3. c) The plane is oblique to the bases and does not intersect them.