## Math 8 Chapter 2 Lesson 4: Area of a trapezoid

## 1. Theoretical Summary

### 1.1. Formula for calculating the area of a trapezoid

First calculate the general formula of a trapezoid, we will have the formula: the average of the 2 bases multiplied by the height between the 2 bases

S = 1/2(a+b) * h

### 1.2. Formula for calculating the area of a parallelogram

The area of a parallelogram is the product of one side times the corresponding altitude \(S = ah\)

## 2. Illustrated exercise

### 2.1. Exercise 1

Calculate the area of trapezoid \(ABED\) according to the given lengths in figure \(140\) and know that the area of rectangle \(ABCD\) is \(828m^2.\)

**Solution guide**

We have \({S_{ABC{\rm{D}}}} = AB.A{\rm{D}} = 828{m^2}\)

\( \Rightarrow AD =\dfrac{828}{AB} =\dfrac{828}{23} = 36 \,(m)\)

Therefore the area of the trapezoid \(ABED\) is:

\({S_{ABED}} = \dfrac{{\left( {AB + DE} \right).AD}}{2} \)\(\,= \dfrac{{\left( {23 + 31} \right).36}}{2} = 972({m^2})\)

### 2.2. Exercise 2

Calculate \(x,\) knowing the polygon in figure \(188\) has an area of \(3375 \,m^2\)

**Solution guide**

The given polygon consists of a trapezoid and a triangle.

Area of trapezoid is \(S_1,\) area of triangle is \(S_2\)

\({S_1} = \dfrac{50+70}{2}.30 = 1800\) (\({m^2}\))

\({S_2} = S – {S_1} = 3375 – 1800 = 1575\) (\({m^2}\))

Again: \({S_2} = \dfrac{1}{2}h.70\)

So the height \(h\) of the triangle is:

\(h = \dfrac{2S_2}{70} = \dfrac{2.1575}{70} = 45\) \((m)\)

Length \(x = 45 + 30 = 75 \,(m)\)

### 2.3. Exercise 3

Given a rectangle \(ABCD\) with side \(AB = 5cm,\, BC = 3cm.\) Draw a parallelogram \(ABEF\) with side \(AB = 5cm\) and whose area is equal to the area. of the rectangle \(ABCD.\) How many such \(ABEF\) can be drawn?

**Solution guide**

On the edge \(CD\) we take any point \(E\) (\(E\) other than \(C\) and \(D\)). From \(A\) draw a line parallel to \(BE\) intersecting the line \(CD\) at \(F\).

The quadrilateral \(ABEF\) has opposite sides parallel, so \(ABEF\) is a parallelogram

We have the area of rectangle \(ABCD\) equal to: \(S_{ABCD}=AB.AD\)

Area of parallelogram \(ABEF\) is equal to: \(S_{ABEF}=AD.EF=AB.AD\) (because \(EF=AB\) do \(ABEF\) is a parallelogram)

Derive: \(S_{ABCD}=S_{ABEF}\)

You can draw an infinite number of such shapes.

## 3. Practice

### 3.1. Essay exercises

**Question 1:** Given a rectangle \(ABCD\) with side \(AB = 5cm,\, BC = 3cm.\) Draw a parallelogram \(ABEF\) with sides \(AB = 5cm,\, BE = 5cm\) and has an area equal to the area of the rectangle \(ABCD.\) How many such shapes \(ABEF\) can you draw?

**Verse 2:** Find the area of a square trapezoid, given that the two bases are \(2\,cm\) and \(4\,cm,\) the angle formed by one side and the great base whose measure is \(45 ^0.\)

**Question 3:** Find the area of a trapezoid, knowing the bases of lengths \(7\,cm\) and \(9\,cm,\) one of the long sides \(8\,cm\) and make with the base a An angle whose measure is \(30°\)

**Question 4: **Prove that every line that passes through the midpoint of the median of the trapezoid and intersects the two bases divides the trapezoid into two trapezoids of equal area.

### 3.2. Multiple choice exercises

**Question 1:** A trapezoid has base lengths 6cm and 4cm respectively and its area is 15cm2. What is the height of the trapezoid?

A. 3cm.

B. 1.5cm

C. 2cm

D. 1cm

**Verse 2: **Given a parallelogram ABCD ( AB//CD ) with AB = CD = 4cm, the height of the parallelogram is h = 2cm. What is the area of a parallelogram?

A. 4(cm^{2} )

B. 8(cm^{2} )

C. 6(cm^{2} )

D. 3(cm^{2} )

**Question 3: **Let ABC be a triangle with BC = 16cm, altitude AH = 8cm. Let M and N be the midpoints of AB and AC respectively. Calculate the area of quadrilateral MNCB?

A. 48cm^{2 }

B. 40cm^{2}

C. 54cm^{2 }

D. 60cm^{2}

**Question 4: **Let ABC be a right triangle at A with AB = 6cm and BC = 10cm . Let M and N be the midpoints of AB and BC respectively. Calculate the area of quadrilateral MNCA?

A. 10 cm^{2 }

B. 12cm^{2}

C. 15cm^{2 }

D. 18cm^{2}

**Question 5: **Let ABC be a triangle with M, N and P being the midpoints of AB, AC and P respectively. The altitudes AH = 10cm and BC = 16cm are known. Calculate the area of quadrilateral MNPB?

A. 20cm^{2 }

B. 30cm2

C. 40cm^{2}

D. 50cm^{2}

## 4. Conclusion

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

- Know the formula for calculating the area of a trapezoid
- Know the formula for calculating the area of a parallelogram

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