3D 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 $AB$.

We know that given two points in space, exactly one line passes through them.

1. a) How many lines can pass through a single point in space?
2. b) How many planes can contain a single line?
3. c) What is the minimum number of non-collinear points needed to define a single plane?

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.

Imagine you have a flat metal piece in the shape of a right-angled trapezium, as shown in the diagram. If we rotate this shape through $360^{\circ}$ around the side measuring $8$ cm (the vertical axis):

1. a) What solid of revolution is generated?
2. b) Draw a sketch of the resulting solid, indicating its basic elements (radii and height).

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

A sports hall has a roof shaped like a regular hexagonal prism, whose top and bottom bases are hexagons and its lateral faces are rectangles.

1. a) Before counting edges or applying any formula, reasonably estimate at first glance whether this solid will have more edges than the sum of its faces and vertices combined, or if the opposite will happen.
2. b) Count in an orderly way the number of faces ($C$), vertices ($V$), and edges ($A$) of the prism and check whether your initial estimate was correct using Euler's formula ($C + V = A + 2$).
3. c) Justify why the total number of edges obtained is reasonable, taking into account how many edges each base contributes and how many vertical edges connect both bases.

An architect is designing an avant-garde monument that is a convex polyhedron. To verify its structure, they need to check Euler's formula. The monument has $8$ hexagonal faces and $6$ square faces.

1. a) Calculate the total number of edges ($A$) of the polyhedron.
2. b) Using Euler's formula, determine the number of vertices ($V$) that the monument will have.

A gift box manufacturer wants to create a special "dual" model. First, they make a regular tetrahedron with edge length $L$. Then, they join the midpoints of all the edges of the tetrahedron to each other.

1. a) What polyhedron is formed inside by joining these midpoints?
2. b) Calculate the number of faces, vertices, and edges of the resulting new polyhedron.
3. c) If the original tetrahedron satisfies Euler's formula ($C + V = A + 2$), show whether the interior polyhedron also satisfies it.

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.

An interior designer wants to install a strand of LED lights across a rectangular room that is $6 \text{ m}$ long, $4 \text{ m}$ wide, and $3 \text{ m}$ high. The cable must go from a corner on the floor ($A$) to the opposite corner on the ceiling ($G$).

1. a) Calculate the minimum length of the cable if it is stretched taut through the interior of the room (the diagonal of the cuboid).
2. b) If the designer decides to run the cable along the walls (without going through the air), what would be the shortest distance between those same two points? (Hint: think about the net of the room).
3. c) State how many edges of the room are parallel to the line containing the height edge where point $A$ is located.

Consider a cheese cube with an edge length of $4 \text{ cm}$. We want to slice it with a very thin knife such that the cross section is a regular hexagon.

1. a) Describe precisely through which points on the edges of the cube the plane must pass to obtain this hexagon.
2. b) How many face diagonals does the cube have in total? And how many space (interior) diagonals?
3. c) If we call the plane containing the hexagon $P$, is there any edge of the cube that is parallel to this plane, or are they all intersecting? Justify your answer.