# Trapezoids

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.

## Test yourself on trapezoid for ninth grade!

True OR False:

In all isosceles trapezoids the bases are equal.

## Types of Trapezoids

There are several types of trapezoids that you will encounter during your studies.
For example, there are trapezoids where the lines look in the same direction and it is called a parallelogram
Additionally, there is a trapezoid that has two sides of the same length and it is called an Isosceles trapezoid. In the first example, we have a right angle trapezoid, which is a trapezoid where the height is equal to the side perpendicular to the bases so that 2 right angles are formed.

## Exercises with Explanations on the Topic: Properties and Types of Trapezoids

### Exercise 1

How do we calculate the area of a trapezoid?

We are given the following isosceles trapezoid with the following characteristics:

What is its height?

Solution:

Formula for the area of a trapezoid:

$\frac{(Base+Base)}{2}\times height$

Substituting these values into our formula we have the following:

$\frac{9+6}{2}\times h=30$

And we solve:

$\frac{15}{2}\times h=30$

$7\frac{1}{2}\times h=30$

$h=\frac{30}{\frac{15}{2}}$

$h=\frac{60}{15}$

$h=4$

Height $BE$ is equal to $4cm$.

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### Exercise 2

Given the area of a trapezoid where its lower base is twice the upper base and $4$ times greater than the height.

The area of the trapezoid is equal to $12 cm²$ (find help starting from $X$)

Calculate the value of $x$.

Solution:

$=\frac{h\times\lparen base+base\rparen}{2}$

$=\frac{x\times(2x+4x)}{2}=\frac{12}{1}$

Given that the lower base is twice the upper base and $4$ times larger than the height.

$24=6x²$ if we divide everything by $6$

$4=x²$ Now taking the square root of both sides $\sqrt{}$

$x=2$

$x=2$

### Exercise 3

Given that $ABCD$ is an isosceles trapezoid with the following characteristics

$AD=AE$

Find the angles of the trapezoid and the angle $\alpha$

Solution:

$∢FAB+∢BAE+∢EAD=180$

$∢72+∢67+∢EAD=180$

$∢EAD=180-72-67=41$

$∢D=∢AED$

This is because $\triangle ADE$ is an isosceles triangle

Opposite the equal sides of the triangle $∆AED$

$∢D=∢AED=\frac{180-∢DAE}{2}=\frac{180-41}{2}=69.5$

$∢D=∢C=69.5$

$∢B=∢\text{BAD}=180-∢D$

The sum of the adjacent angles in the trapezoid are equal to $=180-69.5=110.5$

$180$ between the two bases

Angles opposite by the vertex $∢α=∢B=110.5$

$∢α=∢B=110.5$

Do you know what the answer is?

### Exercise 4

Given that $ABCD$ is an isosceles trapezoid

$BC>BE$

What is the midsegment opposite to $DC$ in the triangle whose base is $DC$ equal to?

Solution:

$\text{AB‖DC}$--->

$∢ABE=∢BEC$

$∢BAE=∢\text{AED}$

Given the figure that: $∢ABE=∢BAE$

$∢BEC=∢\text{AED}$

The rule:

$BE=AE$

Opposite sides are equal in $DABE$, $BE = AE$ is equal to the sides

Side = $\text{BE=AE}$

Side = $\text{BC=AD}$

Angle = $∢BAC=∢AED$

According to the side, side, angle congruence theorem

$△\text{AED}\cong △\text{BEC}$

$\text{EC=ED}$

Corresponding sides between overlapping triangles

$\text{DC=DE+EC=EC+EC=2EC}$

A midsegment in a triangle is equal to half of the side it corresponds to.

Midsegment =

$\frac{1}{2}DC=\frac{1}{2}2EC=EC$

Answer: $EC$

### Exercise 5

The area of the trapezoid $ABCD cm² = X$

The line $AE$ creates the triangle $AED$ and the parallelogram $ABCE$.

It is known that the ratio of the area of the triangle $AED$ to the area of the parallelogram $ABCE$ is $1:3$.

Calculate the ratio between the sides $DE$ and $EC$.

Solution:

To calculate the ratio between the sides we will use the figure that:

$\frac{S∆\text{ADE}}{S∆ABEC}=\frac{1}{3}$

We calculate using the formula to find half of the area and place the ratio.

$S∆ADE=\frac{h\times DE}{2}$

$S=h\times EC$

$\frac{\frac{1}{2}h\cdot DE}{h\cdot EC}=\frac{1}{3}$

To solve the equation, multiply across the factors.

$h\times EC=3(\frac{1}{2}h\times DE)$

$h\times EC=1.5h\times DE$ Dividing everything by $h$ we have the following:

$EC=\frac{\left(1.5h\times DE\right)}{h}$

$EC=1.5DE$

The ratio between $\frac{EC}{DE}$ is $\frac{1}{1.5}$

Answer: $1:1.5$

## Review Questions

What are the characteristics of trapezoids?

The characteristics of a trapezoid are those that help us define or classify which type of trapezoid we are referring to. It's important to remember that there are different types of trapezoids, such as the isosceles trapezoid, the right trapezoid, or a parallelogram. Therefore, it is very important to take into account the characteristics of each one.

What characteristics does a right trapezoid have?

The characteristics of a right trapezoid are that it has two right angles, one acute angle, and one obtuse angle, resulting in two of its sides being equal.

What are the elements of a trapezoid?

As we have already mentioned, a trapezoid is a quadrilateral (4 sides) that has two parallel sides, which are called bases, usually considered as the larger base and the smaller base, it has a height, and it has 4 angles.

What is an isosceles trapezoid?

An isosceles trapezoid is one that has two equal sides, which are the sides that join its parallel lines. This is one of the main characteristics of this type of trapezoid.

Do you think you will be able to solve it?

## examples with solutions for trapezoids

### Exercise #1

The trapezoid ABCD is shown below.

AB = 2.5 cm

DC = 4 cm

Height (h) = 6 cm

Calculate the area of the trapezoid.

### Step-by-Step Solution

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

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

We substitute the given values into the formula:

(2.5+4)*6 =
6.5*6=
39/2 =
19.5

$19\frac{1}{2}$

### Exercise #2

Given the trapezoid:

What is the area?

### Step-by-Step Solution

Formula for the area of a trapezoid:

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

We substitute the data into the formula and solve:

$\frac{9+12}{2}\times5=\frac{21}{2}\times5=\frac{105}{2}=52.5$

52.5

### Exercise #3

Given: $∢C=2x$

$∢A=120°$

isosceles trapezoid.

Find x.

### Step-by-Step Solution

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

$∢C=∢D$

$∢A=∢B$

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

Therefore we can create the formula:

$∢A+∢B+∢C+∢D=360$

We replace according to the existing data:

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

$240+4x=360$

$4x=360-240$

$4x=120$

We divide the two sections by 4:

$\frac{4x}{4}=\frac{120}{4}$

$x=30$

30°

### Exercise #4

What is the perimeter of the trapezoid in the figure?

### Step-by-Step Solution

To find the perimeter we will add all the sides:

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

24

### Exercise #5

True OR False:

In all isosceles trapezoids the bases are equal.

### Step-by-Step Solution

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