## Math 8 Chapter 4 Lesson 1: Rectangle

## 1. Summary of theory

### 1.1. Rectangular

**Define:** A rectangular box is a space shape with 6 faces that are all rectangles.

- A rectangular box has 6 faces, 8 vertices, and 12 edges.
- The two faces facing each other are considered to be the base of the rectangular box, the other faces are called the sides

### 1.2. Cube

A cube is a rectangular box whose 6 faces are all squares.

### 1.3. Plane and line

- Through three noncollinear points define one and only one plane.
- Through two intersecting lines define one and only one plane.
- If a line passes through two distinct points of a plane, then every point of that line is in the plane.

## 2. Illustrated exercise

### 2.1. Exercise 1

Observe the rectangular box \(ABCD.A’B’C’D’\). Name the faces, vertices, and sides of a rectangular box.

**Solution guide**

Faces: \(ABCD, A’B’C’D’, ABB’A’\), \(CDD’C’, ADD’A’, BCC’B’\).

Vertices: \(A, B, C, D, A’, B’, C’, D’\).

Edges: \(AB, BC, CD, DA, A’B’, B’C’, C’D’,\)\(\, D’A’, AA’, BB’, CC’, DD’ \).

### 2.2. Exercise 2

The dimensions of the rectangular box \(ABCD{A_1}{B_1}{C_1}{D_1}\) are \(DC = 5cm, CB = 4cm, BB_1= 3cm\). How many centimeters are the lengths of \(DC_1\) and \(CB_1\)?

**Solution guide**

Since \(ABCD. {A_1}{B_1}{C_1}{D_1}\) is a rectangular box, \(DC{C_1}{D_1};CB{B_1}{C_1}\) is a rectangle.

Derive \(CC_1=BB_1=3cm\)

\(\Delta DC{C_1}\) is square at \(C\), so applying the Pythagorean theorem we have:

\(\eqalign{

& D{C_1} = \sqrt {D{C^2} + C{C_1}^2} \cr

& \,\,\,\,\,\,\,\,\,\,\,\; = \sqrt {{5^2} + {3^2}} = \sqrt {34} \,\,\left( {cm} \right) \cr} \)

\(∆CBB_1\) is square at \(B\), so applying the Pythagorean theorem we have:

\(\eqalign{

& C{B_1} = \sqrt {C{B^2} + B{B_1}^2} \cr

& \,\,\,\,\,\,\,\,\,\,\;\, = \sqrt {{4^2} + {3^2}} = \sqrt {25} = 5( cm) \cr} \)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Fill in the blanks (…)

a. The name of the figure….

b. This image has …..sides

c. This picture has …..face

d. This image has….top

**Verse 2: **See the picture please:

a. Name the planes containing the PR line.

b. Name the planes containing the line PR but not clearly seen in the figure

c. Name the plane that also contains the lines PQ and MV.

**Question 3: **\(ABCD. {A_1}{B_1}{C_1}{D_1}\) is a rectangular box (h. 99)

a) If \(O\) is the midpoint of the segment \({A_1}B\) then \(O\) is the point of the segment \(A{B_1}\) ?

b) \(K\) is a point on edge \(BC\), can \(K\) be a point on edge \(D{D_1}\) ?

**Question 4: **Given the rectangular box \(ABCD. {A_1}{B_1}{C_1}{D_1}\). Draw a diagonal of the face \(DC{C_1}{D_1}\). Will this diagonal intersect the lines \(DC, {D_1}C\), \(D{D_1}\) ?

### 3.2. Multiple choice exercises

**Question 1: **What is the number of faces, vertices, and sides of a cube?

A. 4 faces, 8 vertices, 12 edges.

B. 6 faces, 8 vertices, 12 edges.

C. 6 faces, 12 vertices, 8 edges.

D. 8 faces, 6 vertices, 12 edges.

**Verse 2:** What is the number of pairs of parallel faces in a rectangular box?

A. 2

B. 3

C. 4

D. 5

**Question 3: **Given rectangular box ABCD.A’B’C’D’. Choose the correct statement?

A. ( ABCD ) // ( BCC’B’ )

B. ( BCC’B’ ) // ( ADD’A’ )

C. ( CDD’C’ ) // ( ADD’A’ )

D. ( ABCD ) // ( ADD’A’ )

**Question 4: **Given rectangular box ABCD.A’B’C’D’. Choose the correct statement?

A. AB//CD

B. B’C’//CC’

C. CD//AD

D. BC//BB’

**Question 5: **Which of the following statements is false?

A. Through three noncollinear points define one and only one plane.

B. Through two intersecting lines determine one and only one plane

C. If a line passes through two distinct points of a plane, every point of that line is in the plane.

D. Two parallel planes have at least one point in common.

## 4. Conclusion

Through this lesson, you will learn some key topics as follows:

- Understand the elements of a rectangular box, familiarize yourself with the concepts of straight line points in space, how to sign.
- Know how to determine the number of faces, vertices, and sides of a rectangular box.

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