Space

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

1. a) A stretched headphone cable.
2. b) A flat monitor screen (ignoring its thickness).
3. c) A box where you keep console games.
4. d) A spotlight 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.

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?

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 $ADHE$?

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 centers.
3. c) The plane is oblique to the bases and does not intersect them.

In a design and technology class, students are building a scale model of a modern building. The central piece is a right regular hexagonal prism. To give it an artistic touch, they want to make plane cuts with an electric saw.

1. a) If the cutting plane passes through three non-consecutive vertices of the top base, describe the shape of the resulting cross-section.
2. b) Is it possible to make a cut such that the resulting cross-section is a rectangle? What about a square? Describe how the plane should be in each case.
3. c) If the prism has $12$ base edges and $6$ lateral edges, how many space diagonals (diagonals that do not lie on any face) does this polyhedron have in total?

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.

Imagine you have a flat metal piece in the shape of a right trapezoid, 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).

Given a regular square pyramid, we label its base edges as $b_1, b_2, b_3, b_4$ and its lateral edges as $a_1, a_2, a_3, a_4$. State whether the following statements are true or false, giving reasons:

1. a) Any pair of lateral edges are skew lines.
2. b) A base edge and the lateral edge originating from one of its endpoints are intersecting lines.
3. c) Two opposite base edges are parallel to each other.

Look at the net of this geometric solid. Without performing calculations, answer:

1. a) What polyhedron will be obtained when folding it? Classify it completely.
2. b) How many dihedral angles will the solid have once assembled?
3. c) What is the order of each of its vertices (how many faces meet at them)?

A logistics company uses containers shaped like a right trapezoidal prism (the base is an isosceles trapezium).

1. a) Determine the number of right dihedral angles this solid has.
2. b) If we extend the two non-parallel side faces, explain what would happen: Would they intersect in a line? Would they be parallel? What geometric solid would be formed if we cut that new structure with a plane parallel to the base?
3. c) Check whether a trapezoidal prism with bases of $5$ sides (if it were an irregular pentagonal prism) always satisfies Euler's formula.

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.

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.

An archaeologist has found a piece of crystal in the shape of a perfect regular octahedron. To protect it, they decide to build a wooden box in the shape of a cube that fits the vertices of the octahedron exactly.

1. a) If each edge of the octahedron measures $10 \text{ cm}$, calculate the number of faces, vertices, and edges of the shape and check if Euler's formula holds true.
2. b) Explain with reasons the relationship between the centers of the faces of the wooden cube and the vertices of the octahedron.
3. c) If we cut the octahedron with a plane parallel to one of its faces that passes through the exact center of the solid, what polygon do we get in the cross-section?

We are studying relative positions in space within a complex polyhedron: a regular dodecahedron (12 pentagonal faces).

1. a) Choose any face as plane $\pi$. How many faces of the dodecahedron are parallel to $\pi$?
2. b) How many edges of the dodecahedron are parallel to a given edge?
3. c) If 3 faces meet at each vertex of the dodecahedron, and each face has 5 vertices, work out the total number of vertices of the solid (bearing in mind that each vertex is shared).

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 rectangular prism).
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.