The trapezoid is considered one of the most intimidating shapes for students, therefore we have decided to provide a summary of the general idea behind the trapezoid and explain its properties to them and introduce some types of trapezoids.

Characteristics of the Trapezoid

A trapezoid is a quadrilateral based on 4 sides like any other,
but special in that it will always have two parallel sides also called bases, which we can call the larger base and the smaller base
and it will also have two opposite sides also called legs.

A1 - Characteristics and types of trapezoids


Practice Trapezoid for ninth grade

Exercise #1

Given the trapezoid:

999121212555AAABBBCCCDDDEEE

What is the area?

Video Solution

Step-by-Step Solution

Formula for the area of a trapezoid:

(base+base)2×altura \frac{(base+base)}{2}\times altura

We substitute the data into the formula and solve:

9+122×5=212×5=1052=52.5 \frac{9+12}{2}\times5=\frac{21}{2}\times5=\frac{105}{2}=52.5

Answer

52.5

Exercise #2

The trapezoid ABCD is shown below.

AB = 2.5 cm

DC = 4 cm

Height (h) = 6 cm

Calculate the area of the trapezoid.

2.52.52.5444h=6h=6h=6AAABBBCCCDDD

Video Solution

Step-by-Step Solution

First, let's remind ourselves of the formula for the area of a trapezoid:

A=(Base + Base) h2 A=\frac{\left(Base\text{ }+\text{ Base}\right)\text{ h}}{2}

We substitute the given values into the formula:

(2.5+4)*6 =
6.5*6=
39/2 = 
19.5

Answer

1912 19\frac{1}{2}

Exercise #3

What is the perimeter of the trapezoid in the figure?

444555999666

Video Solution

Step-by-Step Solution

To find the perimeter we will add all the sides:

4+5+9+6=9+9+6=18+6=24 4+5+9+6=9+9+6=18+6=24

Answer

24

Exercise #4

Given: C=2x ∢C=2x

A=120° ∢A=120°

isosceles trapezoid.

Find x.

AAABBBDDDCCC120°2x

Video Solution

Step-by-Step Solution

Given that the trapezoid is isosceles and the angles on both sides are equal, it can be argued that:

C=D ∢C=∢D

A=B ∢A=∢B

We know that the sum of the angles of a quadrilateral is 360 degrees.

Therefore we can create the formula:

A+B+C+D=360 ∢A+∢B+∢C+∢D=360

We replace according to the existing data:

120+120+2x+2x=360 120+120+2x+2x=360

 240+4x=360 240+4x=360

4x=360240 4x=360-240

4x=120 4x=120

We divide the two sections by 4:

4x4=1204 \frac{4x}{4}=\frac{120}{4}

x=30 x=30

Answer

30°

Exercise #5

True OR False:

In all isosceles trapezoids the bases are equal.

Video Solution

Step-by-Step Solution

True: in every isosceles trapezoid the base angles are equal to each other.

Answer

True

Exercise #1

What is the area of the trapezoid in the figure?

777151515222AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

We use the formula to calculate the area of a trapezoid: (base+base) multiplied by the height divided by 2:

(AB+DC)×BE2 \frac{(AB+DC)\times BE}{2}

(7+15)×22=22×22=442=22 \frac{(7+15)\times2}{2}=\frac{22\times2}{2}=\frac{44}{2}=22

Answer

22 22 cm².

Exercise #2

Calculate the area of the trapezoid.

555141414666

Video Solution

Step-by-Step Solution

We use the formula (base+base) multiplied by the height and divided by 2.

Note that we are only provided with one base and it is not possible to determine the size of the other base.

Therefore, the area cannot be calculated.

Answer

Cannot be calculated.

Exercise #3

The trapezoid ABCD is shown below.

AB = 4 cm

DC = 8 cm

Area of the trapezoid (S) = 30 cm²

Calculate the height of the trapezoid.

S=30S=30S=30444888AAABBBCCCDDD

Video Solution

Step-by-Step Solution

We use the formula to calculate the area: (base+base) times the height divided by 2

We replace the existing data:

S=(AB+CD)×h2 S=\frac{(AB+CD)\times h}{2}

30=(4+8)×h2 30=\frac{(4+8)\times h}{2}

We multiply the equation by 2:

60=(4+8)h 60=(4+8)h

60=12h 60=12h

We divide the two sections by 12:

6012=h \frac{60}{12}=h

5=h 5=h

Answer

5

Exercise #4

Look at the trapezoid in the figure.

The long base is 1.5 times longer than the short base.

Find the perimeter of the trapezoid.

222333555

Video Solution

Step-by-Step Solution

First, we calculate the long base from the existing data:

Multiply the short base by 1.5:

5×1.5=7.5 5\times1.5=7.5

Now we will add up all the sides to find the perimeter:

2+5+3+7.5=7+3+7.5=10+7.5=17.5 2+5+3+7.5=7+3+7.5=10+7.5=17.5

Answer

17.5

Exercise #5

The perimeter of the trapezoid in the diagram is 25 cm. Calculate the missing side.

777444111111

Video Solution

Step-by-Step Solution

We replace the data in the formula to find the perimeter:

25=4+7+11+x 25=4+7+11+x

25=22+x 25=22+x

2522=x 25-22=x

3=x 3=x

Answer

3 3 cm

Exercise #1

In an isosceles trapezoid ABCD

B=3x ∢B=3x

D=x ∢D=x


Calculate the size of angle B ∢B .

Video Solution

Step-by-Step Solution

To answer the question, we must know an important rule about isosceles trapezoids:

The sum of the angles that define each of the trapezoidal sides (not the bases) is equal to 180

Therefore:

∢B+∢D=180

3X+X=180

4X=180

X=45

It's important to remember that this is still not the solution, because we were asked for angle B,

Therefore:

3*45 = 135

And this is the solution!

Answer

135°

Exercise #2

Do the diagonals of the trapezoid necessarily bisect each other?

Step-by-Step Solution

The diagonals of an isosceles trapezoid are always equal to each other,

but they do not necessarily bisect each other.

(Reminder, "bisect" means that they meet exactly in the middle, meaning they are cut into two equal parts, two halves)

For example, the following trapezoid ABCD, which is isosceles, is drawn.

Using a computer program we calculate the center of the two diagonals,

And we see that the center points are not G, but the points E and F.

This means that the diagonals do not bisect.

 

 

Answer

No

Exercise #3

The perimeter of the trapezoid equals 22 cm.

AB = 7 cm

AC = 3 cm

BD = 3 cm

What is the length of side CD?

AAABBBDDDCCC733

Video Solution

Step-by-Step Solution

Since we are given the perimeter of the trapezoid and not the length of CD, we can calculate:

22=3+3+7+CD 22=3+3+7+CD

22=CD+13 22=CD+13

2213=CD 22-13=CD

9=CD 9=CD

Answer

9

Exercise #4

What is the area of the trapezoid in the figure?

222999777AAABBBCCCDDD

Video Solution

Step-by-Step Solution

We use the formula: (base + base) multiplied by height divided by 2:

S=(AB+DC)×h2 S=\frac{(AB+DC)\times h}{2}

Keep in mind that AD is the height of a trapezoid:

We replace the existing data in the formula:

S=(2+9)×72 S=\frac{(2+9)\times7}{2}

S=11×72=772=38.5 S=\frac{11\times7}{2}=\frac{77}{2}=38.5

Answer

38.5 38.5 cm².

Exercise #5

Given the trapezoid in front of you:

AAABBBCCCDDD151269

Given h=9, DC=15.

Since the area of the trapezoid ABCD is equal to 126.

Find the length of the side AB.

Video Solution

Step-by-Step Solution

We use the formula to calculate the area: (base+base) times the height divided by 2

S=(AB+CD)×h2 S=\frac{(AB+CD)\times h}{2}

We input the data we are given:

126=(AB+15)×92 126=\frac{(AB+15)\times9}{2}

We multiply the equation by 2:

252=9AB+135 252=9AB+135

252135=9AB 252-135=9AB

117=9AB 117=9AB

We divide the two sections by 9

13=AB 13=AB

Answer

13