# What is a box – Troubleinthepeace

Figure 33. Prism shape.

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The vertices A1, A2, A3, A4, A5 can move freely on the plane P.Ta has the following observations:

The quadrilateral domains A1A2A’2A’1, A2A3A’3A’2, . . . AnA1A’1A’n are all parallelogram regions, or simply called parallelograms.

Two polygons A1A2. . An and A’1A’2. . A’n has equal and parallel sides, respectively.

Define: Formed by parallelograms A1A2A’2A’1, A2A3A’3A’2, . . . AnA1A’1A’n and two polygon domains A1A2. . . An’A’1A’2. . . A’n is called a prism (or prism for short).

These parallelograms are called the side faces of the prism. Two polygon domains A1A2. . An; A’1A’2. . . A’n is called the two bases of the prism

Line segments A1A’1, A2A’2, . . . AnA’n is called the lateral edges of the prism. These side segments are all equal and parallel to each other.

The vertices of the two base polygons are called the vertices of the prism. The prism as above is denoted as prism A1A2. . An. A’1A’2. . An.

If the base of the prism is a triangle, quadrilateral, pentagon, . . then the corresponding prism is called triangular prism, quadrangular prism, pentagonal prism, . . .

Figure 34a. Triangular prism

It is possible to shift the vertices of this prism.

It is possible to shift the vertices of this prism.

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Figure 34c. The shape of the pentagon

It is possible to shift the vertices of this prism.

2. Box shape

Define: A quadrilateral whose base is a parallelogram is called a box.

Thus, a box with four sides and two bases are parallelograms

Figure 35. Box shape

It is possible to shift the vertices of this box.

Two faces that are parallel to each other are called opposite faces. Any two opposite faces can be used as the bottom faces of a box

A box has three pairs of opposite faces, and any two opposite faces are equal.

A box has 8 vertices and 12 edges (including the side and bottom edges).

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The edges are divided into three groups, each group consists of four parallel and equal sides.

Two vertices of a box are said to be opposite vertices if they do not belong to the same face. Example ABCD box. A’B’C’D’ we have pairs of opposite vertices A and C’, B and D’, C and A’, D and B’.

The line segment joining two opposite vertices is called the diagonal of the box. Each box has four diagonals.

Two sides are said to be opposite if they are parallel but not on the same face of the box.

The diagonal of the box is a parallelogram whose two sides are opposite sides of the box. Each box has 6 diagonal faces.

The diagonals of each of these diagonals are diagonals of the box.

Thus, in each box, four diagonals intersect at the midpoint of each line. That intersection point is called the center of the box. The center of the box is also the center of the diagonals

QUESTIONS AND EXERCISES

1. Given a triangular prism ABC. A’B’C’ with sides AA’, BB’, CC’. Let M and M’ be the midpoints of sides BC and B’C’ respectively.

a) Prove that AM//AM’

b) Find the intersection of the plane (AB’C’) with the line A’M

c) Find the intersection d of two planes (AB’C’) and (BA’C’)

d) Find the intersection G of the line d with the plane (AMA’). Prove that G is the centroid of triangle AB’C’.

2. Given the box ABCD. A’B’C’D’ has sides AA’, BB’, CC’, DD’.

a) Prove that two planes (BDA’) and (B’D’C) are parallel.

b) Prove that the diagonal AC’ passes through the centroids G1 and G2 of the two triangles BDA’ and B’D’C.

c) Prove that G1 and G2 divide the line segment AC’ into three equal parts.

d) Let O, I be the centers of parallelograms ABCD and AA’C’C, respectively. Determine the cross-section of the plane (A’IO) with the given box.

3. Prove that the six midpoints of the sides AB, AD, DD’, D’C’, C’B’, B’B of the box ABCD. A’B’C’D’ (with AA’ // BB’ // CC’// DD’) lies on a plane. Prove that the plane is parallel to the plane (AB’D’).

4. Prove that the sum of the squares of all the diagonals of the box is equal to the sum of the squares of all the sides of the box.

5. Given a triangular prism ABC. A’B’C’ has AA’//BB’//CC’. Let H be the mid point of side A’B’.

a) Prove that the line CB’ is parallel to the plane (AHC’).

b) Find the intersection d of the two planes (AB’C’) and (A’BC). Prove that d is parallel to the plane (BB’C’C)

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