Types of trapezoids

Properties of a regular trapezoid
• A quadrilateral with only 2 parallel sides.
• Angles resting on the same leg are supplementary to 180 degrees, so the sum of all angles is 360 degrees.
• The diagonal of the trapezoid creates equal alternate angles between parallel lines.

Properties of a trapezoid that is a parallelogram
• A quadrilateral with 2 pairs of parallel sides – parallel bases and parallel legs.
• Its opposite sides are equal.
• Its opposite angles are equal.
• The diagonals bisect each other.

Properties of an Isosceles Trapezoid
• A quadrilateral with one pair of parallel sides and another pair of non-parallel but equal sides.
• The base angles are equal.
• The diagonals are equal.

Properties of a Right-Angled Trapezoid
• A quadrilateral with only one pair of parallel sides and 2 angles each equal to 90 degrees.
• The height of the trapezoid is the leg on which the two right angles rest.
• The other 2 angles add up to 180 degrees.

Types of Trapezoids

Practice Trapeze

Examples with solutions for Trapeze

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

Look at the trapezoid in the diagram.

101010777121212777

What is its perimeter?

Video Solution

Step-by-Step Solution

In order to calculate the perimeter of the trapezoid we must add together the measurements of all of its sides:

7+10+7+12 =

36

And that's the solution!

Answer

36

Exercise #3

Given the trapezoid:

444999666131313

What is its perimeter?

Video Solution

Step-by-Step Solution

The problem requires calculating the perimeter of the trapezoid by summing the lengths of its sides. Based on the given trapezoid diagram, the side lengths are clearly marked as follows:

  • First side: 4 4
  • Second side: 9 9
  • Third side: 6 6
  • Fourth side: 13 13

According to the formula for the perimeter of a trapezoid:

P=a+b+c+d P = a + b + c + d

Substituting the respective values:

P=4+9+6+13 P = 4 + 9 + 6 + 13

Calculating the sum, we find:

P=32 P = 32

Thus, the perimeter of the trapezoid is 32 32 .

Answer

32

Exercise #4

The trapezoid ABCD is shown below.

Base AB = 6 cm

Base DC = 10 cm

Height (h) = 5 cm

Calculate the area of the trapezoid.

666101010h=5h=5h=5AAABBBCCCDDD

Video Solution

Step-by-Step Solution

First, we need to remind ourselves of how to work out the area of a trapezoid:

Formula for calculating trapezoid area

Now let's substitute the given data into the formula:

(10+6)*5 =
2

Let's start with the upper part of the equation:

16*5 = 80

80/2 = 40

Answer

40 cm²

Exercise #5

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 #6

The trapezoid ABCD is shown below.

AB = 5 cm

DC = 9 cm

Height (h) = 7 cm

Calculate the area of the trapezoid.

555999h=7h=7h=7AAABBBCCCDDD

Video Solution

Step-by-Step Solution

The formula for the area of a trapezoid is:

Area=12×(Base1+Base2)×Height \text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}

We are given the following dimensions:

  • Base AB=5AB = 5 cm
  • Base DC=9DC = 9 cm
  • Height h=7h = 7 cm

Substituting these values into the formula, we have:

Area=12×(5+9)×7 \text{Area} = \frac{1}{2} \times (5 + 9) \times 7

First, add the lengths of the bases:

5+9=14 5 + 9 = 14

Now substitute back into the formula:

Area=12×14×7 \text{Area} = \frac{1}{2} \times 14 \times 7

Calculate the multiplication:

12×14=7 \frac{1}{2} \times 14 = 7

Then multiply by the height:

7×7=49 7 \times 7 = 49

Thus, the area of the trapezoid is 49 cm2^2.

Answer

49 cm

Exercise #7

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 #8

What is the perimeter of the trapezoid in the figure?

7.57.57.54441.51.51.5333

Video Solution

Step-by-Step Solution

To find the perimeter of the trapezoid, we will sum the lengths of all its sides. The given side lengths are:

  • Base 1: 7.5 7.5
  • Base 2: 1.5 1.5
  • Leg 1: 3 3
  • Leg 2: 4 4

Using the formula for the perimeter P P of the trapezoid, we have:

P=a+b+c+d P = a + b + c + d

Substituting in the given values:

P=7.5+1.5+3+4 P = 7.5 + 1.5 + 3 + 4

Performing the addition:

P=7.5+1.5=9 P = 7.5 + 1.5 = 9

P=9+3=12 P = 9 + 3 = 12

P=12+4=16 P = 12 + 4 = 16

Therefore, the perimeter of the trapezoid is 16 16 .

Answer

16

Exercise #9

Look at the trapezoid in the figure.

Calculate its perimeter.

2.52.52.510.410.410.45.35.35.3666

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify all given side lengths of the trapezoid.
  • Step 2: Apply the formula for the perimeter of the trapezoid.
  • Step 3: Sum up the lengths to find the perimeter.

Now, let's work through each step:
Step 1: The problem gives us the lengths of the trapezoid's sides:
- AB=2.5 AB = 2.5
- BC=10.4 BC = 10.4
- CD=5.3 CD = 5.3
- DA=6 DA = 6

Step 2: We use the formula for the perimeter of a trapezoid:

P=AB+BC+CD+DA P = AB + BC + CD + DA

Step 3: Plugging in the given values, we calculate:

P=2.5+10.4+5.3+6 P = 2.5 + 10.4 + 5.3 + 6

Calculating further, we have:

P=24.2 P = 24.2

Therefore, the perimeter of the trapezoid is 24.2 24.2 .

Answer

24.2

Exercise #10

Below is an isosceles trapezoid

If D=50° ∢D=50°

Determine the value of B ∢B ?

AAABBBDDDCCC50°

Video Solution

Step-by-Step Solution

Let's recall that in an isosceles trapezoid, the sum of the two angles on each of the trapezoid's legs equals 180 degrees.

In other words:

A+C=180 A+C=180

B+D=180 B+D=180

Since angle D is known to us, we can calculate:

18050=B 180-50=B

130=B 130=B

Answer

130°

Exercise #11

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 #12

True OR False:

In all isosceles trapezoids the base Angles are equal.

Video Solution

Step-by-Step Solution

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

Answer

True

Exercise #13

Do isosceles trapezoids have two pairs of parallel sides?

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Define the geometric properties of a trapezoid.
  • Step 2: Define the geometric properties of an isosceles trapezoid.
  • Step 3: Conclude whether an isosceles trapezoid has two pairs of parallel sides based on these definitions.

Now, let's work through each step:
Step 1: A trapezoid is defined as a quadrilateral with at least one pair of parallel sides.
Step 2: An isosceles trapezoid is a special type of trapezoid where the non-parallel sides (legs) are of equal length. Its defining feature is having exactly one pair of parallel sides, which is the same characteristic as a general trapezoid.
Step 3: Since the definition of a trapezoid inherently allows for only one pair of parallel sides, an isosceles trapezoid, as a type of trapezoid, cannot have two pairs of parallel sides. A quadrilateral with two pairs of parallel sides is typically designated as a parallelogram, not a trapezoid.

Therefore, the solution to the problem is that isosceles trapezoids do not have two pairs of parallel sides. No.

Answer

No

Exercise #14

What is the area of the trapezoid in the diagram below?

777333AAABBBCCCDDDEEEFFF4

Video Solution

Step-by-Step Solution

To determine the area of the trapezoid, we will follow these steps:

  • Step 1: Identify the provided dimensions of the trapezoid.
  • Step 2: Apply the formula for the area of a trapezoid.
  • Step 3: Perform the arithmetic to calculate the area.

Let's proceed through these steps:

Step 1: Identify the dimensions
The given dimensions from the diagram are:
Height h=3 h = 3 cm.
One base b1=4 b_1 = 4 cm.
The other base b2=7 b_2 = 7 cm.

Step 2: Apply the area formula
To find the area A A of the trapezoid, use the formula:
A=12×(b1+b2)×h A = \frac{1}{2} \times (b_1 + b_2) \times h

Step 3: Calculation
Substituting the known values into the formula:
A=12×(4+7)×3 A = \frac{1}{2} \times (4 + 7) \times 3

Simplify the expression:
A=12×11×3 A = \frac{1}{2} \times 11 \times 3

Calculate the result:
A=12×33=332=16.5 A = \frac{1}{2} \times 33 = \frac{33}{2} = 16.5 cm²

The area of the trapezoid is therefore 16.5 16.5 cm².

Given the choices, this corresponds to choice : 16.5 16.5 cm².

Therefore, the correct solution to the problem is 16.5 16.5 cm².

Answer

16.5 16.5 cm²

Exercise #15

What is the area of the trapezoid in the diagram?

555138

Video Solution

Step-by-Step Solution

To find the area of the trapezoid, we will follow these steps:

  • Step 1: Identify the given dimensions of the trapezoid.
  • Step 2: Apply the area formula for a trapezoid using these dimensions.
  • Step 3: Perform the calculation to determine the area.

Let's work through each step more clearly:
Step 1: From the problem, we identify that the trapezoid has one base b1=13b_1 = 13 units, another base b2=8b_2 = 8 units, and its height h=5h = 5 units.
Step 2: The formula for the area of a trapezoid is:

A=12×(b1+b2)×h A = \frac{1}{2} \times (b_1 + b_2) \times h

Step 3: Substitute the values into the formula:

A=12×(13+8)×5 A = \frac{1}{2} \times (13 + 8) \times 5

A=12×21×5 A = \frac{1}{2} \times 21 \times 5

A=12×105 A = \frac{1}{2} \times 105

A=52.5units2 A = 52.5 \, \text{units}^2

Therefore, the area of the trapezoid is 52.5units2 52.5 \, \text{units}^2 .

Answer

52.5 52.5 cm²