## Math 8 Chapter 4 Lesson 4: Vertical Prism

## 1. Theoretical Summary

### 1.1. Basic knowledge

The figure on the side is called a vertical prism. In this picture:

+ \(A, B, C, D, {A_1},{B_1},{C_1},{D_1}\) are vertices.

\(AB{B_1}{A_1},BC{C_1}{B_1}\)… are rectangles, called side faces.

+ \(A{A_1};B{B_1};C{C_1};D{D_1}\) are parallel to each other and equal, they are called lateral edges.

+ Two sides \(ABCD\) and \({A_1}{B_1}{C_1}{D_1}\) are two bases. The upper prism has two bases that are quadrilaterals, so it is called a quadrangular cylinder, symbol : \(ABCD. {A_1}{B_1}{C_1}{D_1}\)

### 1.2. Attention

A rectangular box, a cube is also a vertical prism.

A vertical prism whose base is a parallelogram is called a vertical box.

## 2. Illustrated exercise

### 2.1. Exercise 1

The figure shows a vertical prism whose base is a triangle. Which of the following statements is correct?

a) Sides AB and AD are perpendicular to each other;

b) The sides BE and EF are perpendicular to each other;

c) Sides AC and DF are perpendicular to each other

d) The sides AC and DF are parallel to each other;

e) Two planes (ABC) and (DEF) parallel to each other;

f) Two planes (ACFD) and (BCFE) are parallel to each other;

g) Two planes (ABED) and (DEF) are perpendicular to each other.

__Solution guide__

a) Wrong because \(AB\) is not a side edge.

b) Wrong because \(EF\) is not a side edge.

c) Wrong because \(AC\) and \(DF\) are not lateral and are not perpendicular.

d) Wrong because \(AC\) and \(DF\) are not side edges.

e) True because \(mp(ABC) // mp(DEF)\).

g) Wrong because \(mp(ACFD)\) and \(mp(BCFE)\) intersect in a straight line \(CF\).

h) True because in a vertical prism, the sides and base are perpendicular to each other.

### 2.2. Exercise 2

An equilateral triangular prism has base a and height h. Calculate \({S_{xq}},\,{S_{tp}}\) and V of the prism.

__Solution guide__

An equilateral triangle prism is a vertical prism whose base is an equilateral triangle.

Let H be the mid point of BC.

\(\Delta ABC\) are: \(HB = \frac{1}{2}BC = \frac{1}{2}a\)

\(\Delta AHB\) square at H: \(A{H^2} = AB – B{H^2} = {a^2} – {\left( {\frac{a}{2}} \ right)^2} = \frac{{3{a^2}}}{4}\)

\( \Rightarrow AH = \frac{{a\sqrt 3 }}{2} \Rightarrow B = {S_{ABC}} = \frac{1}{2}BC.AH = \frac{{{a^2 }\sqrt 3 }}{4}\)

We have: \({S_{xq}} = 3.AB.AA’ = 3a.h\)

\({S_{tp}} = {S_{xq}} + 2{S_{day}} = 3ah + 2\frac{{{a^2}\sqrt 3 }}{4} = a\left( { { \frac{{h + a\sqrt 3 }}{4}} \right)\)

\(V = Bh = \frac{{{a^2}\sqrt 3 }}{4}.h = \frac{{{a^2}h\sqrt 3 }}{4}.\)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Please indicate:

a) If a prism has six faces, what shape is the base of the prism?

b) If a vertical prism has eight faces, what shape is the base of the prism?

**Verse 2: **\(ABCD.XYHK\) is a vertical prism, the base is a rectangle (h.120)

a) Observe the figure and point out pairs of parallel planes.

b) Which pairs of planes are perpendicular to each other?

c) Are the two faces \((BCHY)\) and \((KXYH)\) perpendicular to each other?

**Question 3: **People dig a ditch 20m long, 1.5m deep. On the surface has a length of 1.8m and the bottom of the ditch is 1.2m

1. Calculate the volume of the soil mass to be excavated.

2. People move the soil mass to spread it on a rectangular piece of land with dimensions of 30 x 60m. The amount of land is transferred by a car that can carry each \(6{m^3}\) of land. Ask:

a) What is the thickness of the layer of soil spread over the plot?

b) Number of trips by car to load the entire block of earth.

**Question 4: **Are the two planes containing the bases of a vertical prism parallel?

– Are the sides perpendicular to the two base planes?

– Are the side faces perpendicular to the two bottom planes?

### 3.2. Multiple choice exercises

**Question 1:** If a prism is vertical, the base is a triangle, then the prism has:

A. \(6\) face, \(9\) edge, \(5\) vertex

B. \(5\) face, \(9\) edge, \(6\) vertex

C. \(6\) face, \(5\) edge, \(9\) vertex

D. \(5\) face, \(6\) edge, \(9\) vertex.

**Verse 2: **Choose the correct statement from the following statements:

A. A triangular prism has 4 faces and 6 vertices.

B. A triangular prism has 5 faces and 6 vertices.

C. Triangular prism has 4 faces, 5 vertices

D. A triangular prism has 4 faces and 4 vertices.

**Question 3: **Let ABC.A’B’C’ vertical prism, AB = 6cm; AC = 8cm, AA’ = 5cm and surrounding area is 120cm2. What is triangle ABC?

A. Isosceles triangle

B. Sharp triangle

C. Obtuse triangle

D. Right triangle

**Question 4: **Consider a triangular prism whose base lengths are 4 cm, 6 cm, and 8 cm. If the area around is 90cm^{2}. Calculate the height of the prism?

A. 5cm

B. 6cm

C. 4cm

D. 8cm

**Question 5: **Given a vertical prism whose base is a regular hexagon of side 6cm and height of the prism is 6cm. Calculate the perimeter of the prism?

A. 160cm^{2}

B. 216cm^{2}

C. 250cm^{2}

D. 320cm^{2}

## 4. Conclusion

Through this lesson, you will learn some of the main topics as follows:

- Understand the concept of a vertical prism, and its elements
- Vertical prism recognition, side face recognition, bottom face recognition, naming, drawing

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