Lines, Planes, and Solids in 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 (excluding its thickness).
3. c) A box where you keep your console games.
4. d) A point of light on the ceiling.

We know that a single line passes through two points in space.

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?

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 position:

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

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

Observe the following cube or regular hexahedron whose vertices have been named. Identify the relative positions of the indicated edges:

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

Given a regular square-based pyramid, we name its base edges $b_1, b_2, b_3, b_4$ and its lateral edges $a_1, a_2, a_3, a_4$. State whether the following statements are true or false, justifying your answers:

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

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

If we draw a straight line passing through the midpoints of the edges $AE$ and $CG$, and another straight line joining 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 the points $B$, $D$ and $H$, what type of cross section (plane figure) is formed when cutting the cube?
3. c) How many edges of the cube intersect the plane defined by the face $ADHE$?

Draw in your notebook (or describe) the situation 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 is intersecting the plane (think of a pencil piercing a sheet of paper).

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

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

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

1. a) The plane of the floor and the plane of the ceiling.
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 plane of the floor.

In the design of an advertising tent in the shape of a right triangular prism, we are interested in analyzing the relationship between its planes. If we consider the plane $\pi$ containing the lower base of the prism:

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

A cut is made to a right cylinder of radius $r = 4$ in and height $h = 10$ in by a plane. Describe what plane figure 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 center.
3. c) The plane is oblique to the bases and does not intersect them.

Consider a cube of cheese with a side of $4 \text{ cm}$. We want to cut it with a very thin knife so 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 diagonals (internal)?
3. c) If we call $P$ the plane containing the hexagon, indicate one edge of the cube that is parallel to this plane and one that intersects it.

In a design and technology class, they 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 upper base, describe the shape of the cross-section obtained.
2. b) Is it possible to make a cut in such a way that the resulting cross-section is a rectangle? And a square? Describe what the plane should be like in each case.
3. c) If the prism has $12$ base edges and $6$ lateral edges, how many space diagonals (that do not lie on the faces) does this polyhedron have in total?

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

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

An architect is designing an avant-garde monument that is a convex polyhedron. To verify its structure, he needs 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 ($V$) the monument will have.

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

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

A gift box manufacturer wants to create a special "dual" model. First, he makes a regular tetrahedron with an edge of $L$. Then, he joins 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 new resulting polyhedron.
3. c) If the original tetrahedron satisfies Euler's formula ($C + V = A + 2$), prove whether the inner polyhedron also satisfies it.

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

1. a) What polyhedron will be obtained when folding it? Classify it fully.
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 in the shape of a right trapezoidal prism (the base is an isosceles trapezoid).

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

We are studying the 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, reason by calculation the total number of vertices of the solid (taking into account that each vertex is shared).

Many geometric solids are generated by rotating a 2D shape around an axis. These are called solids of revolution. Identify which shape (rectangle, right-angled triangle or semicircle) has generated each of these objects when rotated:

1. a) A cylindrical soft drink can.
2. b) A traffic cone.
3. c) A bocce ball.

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

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

An interior designer wants to place an LED light cable crossing a rectangular room $6 \text{ m}$ long, $4 \text{ m}$ wide and $3 \text{ m}$ high. The cable must go from one corner of the floor ($A$) to the opposite corner of the ceiling ($G$).

1. a) Calculate the minimum length of the cable if it must be stretched through the inside of the room (the space diagonal of the cuboid).
2. b) If the designer decides to run the cable along the walls (without crossing 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.