# Rectangle

## Properties of a Rectangle

A rectangle is a quadrilateral with two pairs of parallel opposite edges (sides), the angles of which all equal 90 degrees.

1. The pairs of sides in a rectangle are opposite, equal, and parallel.
2. Each of the angles in a rectangle are equal to 90 degrees.
3. The diagonals of a rectangle are equal.
4. The diagonals of a rectangle intersect and do so at the midpoint of each other.
5. Since the diagonals are equal, so are their halves.

Note:
The diagonals of a rectangle are not perpendicular (they are oblique) and do not cross the angles of the rectangle.

## Test yourself on rectangle for ninth grade!

True or false?

One of the angles in a rectangle may be an acute angle.

## Definition of a Rectangle

A rectangle is a parallelogram that has angles each measuring 90º (right angles).
Since a rectangle is a type of parallelogram, it has all of the the properties of a parallelogram.

## Demonstration of a Rectangle

If a parallelogram has an angle of 90º or is a parallelogram with diagonals of the same length, then it is therefore a rectangle.

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## The Rectangle

The rectangle is an interesting figure that has certain special characteristics. Once we have become familiar with its specific properties, we will be able to quickly determine whether or not any figure we are presented with is in fact a rectangle.

## What is a Rectangle?

A rectangle is a quadrilateral with angles measuring 90 degrees each.

If you are interested in this article, you may also be interested in the following articles:

Area of the rectangle

The perimeter of a rectangle

Rectangles of Equivalent Area and Perimeter

Area of a right triangle

Congruence of right triangles (in the context of the Pythagorean Theorem)

From a quadrilateral to a rectangle

In Tutorela website, you will find a variety of other helpful articles about mathematics!

Do you know what the answer is?

## Properties of the Rectangle

• The pairs of sides are opposite, equal, and parallel.

For example:
$AB=DC$
$AB∥DC$

and also:

$AD=BC$
$AD∥BC$

• All angles of a rectangle measure 90 degrees each.

For example:
$∢A=∢B=∢C=∢D=90$

Let's look at a basic exercise that includes a rectangle:

Rectangle

Shown above is the rectangle $ABDC$, where $AB=6$ and $BD=3$.

We must find the length of the sides $AC$ and $CD$.

According to the properties we have learned about rectangles, their opposite sides have the same length.

Therefore, the following is true:

$AB=CD= 6$

$AC=BD=3$

## Diagonals in the Rectangle

Now let's move on to the properties of the diagonals of the rectangle
Let's redraw our rectangle without any labels so that we don't get confused:

Diagonals of the Rectangle

• The diagonals of the rectangle are equal.

That is, $AC=BD$.

Useful information:
You can verify this fact easily by means of the congruence of triangles $⊿ADC$ and $⊿BCD$
according to LAL.

 Argumento Explicación Lado $AD=BC$ Según las propiedades del rectángulo cada par de lados opuestos son iguales. Ángulo $∢ADC=∢BCD$ Según las propiedades del rectángulo todos sus ángulos son iguales. Lado $DC=DC$ Lados opuestos, tienen la misma longitud. De esto se desprende que:$⊿BCD=⊿ADC$ Según LAL. Entonces:$AC=BD$ Según congruencia de ángulos.
• The diagonals of the rectangle intersect. Not only do they intersect, but they do so at the midpoint of each other. Since the diagonals are equal, so are their halves.

In other words:
$AE=BE=CE=DE$

Important!
The diagonals of a rectangle are not perpendicular!
They do not form a right angle with each other.

Also:
The diagonals do not cross the angles of the rectangle.

Note:
Since a rectangle is a type of parallelogram, it has all the properties of a parallelogram.

How can we prove that a certain figure is a rectangle?
The technical definition of a rectangle is:
A parallelogram that contains a right angle (90º).

Also,
A parallelogram with diagonals of the same length is a rectangle.
A rectangle is a type of parallelogram with special properties.
If we are presented with a parallelogram with an angle of 90º or one with diagonals of the same length, we can safely determine that it is a rectangle!

How can we prove that a quadrilateral is, in fact, a rectangle?
There are two ways:

### The First Method: Checking the Angles

Check if three of the angles measure 90º.
If they do, then the quadrilateral is a rectangle.
(We do not need to measure the fourth angle since, as all angles must add up to 360º, it will also measure 90º).

Do you think you will be able to solve it?

### The Second Method: Demonstration of parallelogram, then, of rectangle.

See if you can prove that the given quadrilateral is a parallelogram.
Don't know exactly how to prove it? Then have a look at our guide "From quadrilateral to parallelogram" to learn how to identify a parallelogram from 20 km away!

Once you have, you will know that a parallelogram is in fact a rectangle based on these rules:

• If the parallelogram has an angle of 90º, then it is a rectangle.
• If the diagonals of the parallelogram are equal, then it is a rectangle.

You can prove that a given parallelogram is a rectangle by applying any of these rules:

• If the parallelogram has an angle of 90º, then it is a rectangle.
• If the diagonals of the parallelogram are equal, then it is a rectangle.

Note:
The technical definition of a rectangle is: a parallelogram with a right angle (90º).
Therefore, if we have a parallelogram that has an angle of 90 degrees, we can determine that it is a rectangle.
Furthermore, another property of a rectangle is that its diagonals are of the same length.
Remembering the second rule, we can check the diagonals.
If they are equal, then we have proven that it is a rectangle.

Great! Now you know everything you need to know about rectangles.

## Rectangle Exercises

### Exercise 1

Have a look at the two rectangles in the figure:

Question:

What is the area of the blank area?

Solution:

To answer the question, we subtract the area of the small rectangle from the area of the large rectangle.

The formula for calculating a rectangular area is base multiplied by the height.

Let's start by calculating the area of the large rectangle.

The base of the large rectangle consists of a side $DC=DG+GC$.

$DC=4+5$

$DC=9$ (base)

The height of the large rectangle is: $DA=DE+EA$.

$AD=2+2=4$

Therefore, $DA=4$.

Now to calculate the area of the large rectangle:

$DC\times DA=9\times4=36cm²$

In the second step, we calculate the area of the small rectangle. The formula is the same: base multiplied by the height.

The area of the small rectangle is equal to $DG\times DE$.

Therefore, the area of the small rectangle is equal to $2\times4=8cm²$.

Now all we have left to do is to calculate the area of the large rectangle minus the area of the small rectangle, leaving us with the area of the blank section.

$36-8=28$

The blank area in the figure is $28$ cm².

### Exercise 2

Below is an orthohedron with the dimensions shown in the figure.

Question:

Which rectangles make up the orthohedron?

Solution:

Based on the fact that an orthohedron is a symmetric box—meaning that the corner angles are always 90 degrees—it can be concluded that there are $3$ pairs of equal rectangles in the orthohedron.

First pair: a rectangle with a base of $5$ cm and a height of $3$ cm.

Second pair: a rectangle with a base of $6$ cm and a height of $3$ cm.

Third pair: a rectangle with a base of $6$ cm and a height of $5$ cm.

### Exercise 3

Below is the rectangle $ABCD$ with an area of $42$ cm², where $AD$ is equal to $12$ cm.

Question:

What is the value of the side $DC$?

Solution:

To answer this question we will first look at the data we have:

The height ($AD$) of the rectangle is $12$.

We will call the base $X$.

The area of the rectangle $ABCD=S=42cm²$.

To answer the question, we substitute the values we have in to the formula to calculate the area of the rectangle: height * base = area of the rectangle.

Next, we present the data in a formula to calculate the area of a rectangle:

$12\times X=42$

Finally, divide the equation by $12$.

$X=3.5$

$AB=DC=3.5\operatorname{cm}$

Do you know what the answer is?

### Exercise 4

Below is a rectangle comprised of two squares:

Question:

What is the area of the rectangle?

Solution:

Since the two squares are identical and adjacent, they form a rectangle divided in the middle.

Next, we need to remember the properties of a square. One of the properties of a square is that all of its sides are equal—that is, the side and the height of the square are equal to $5$.

The calculation can be performed in several ways. However, in this case we will answer the question by calculating the area of a square and multiplying it by $2$.

The formula for calculating a square area is squaring a side:

$\left(5²\right)$

Then all that is left to do is to multiply by $2$.

Therefore, the area of a square is $25\times2=50$.

This is because the rectangle is made up of $2$ identical squares, which means it is equal to $50$ cm².

$50cm²$

### Exercise 5

The area of the rectangle is equal to $256$ cm².

One side is $4$ times longer than the other.

Question:

What are the dimensions of the rectangle?

Solution:

We will substitute the given values into the formula used to calculate the area of the rectangle:

Height * base = area of the rectangle.

$X\times4X=256$

$4X²=256$

Divide the equation by $4$ to get $X²=64$.

Take the square root of both sides of the equation to solve $X$.

$X=8$

Now all you have to do is place $X=8$ in the figure.

Since a rectangle is a symmetric shape, we can easily work out the dimensions of all the sides of the rectangle.

All sides of the rectangle:

## Review Questions

### What is a Rectangle?

It is a geometric figure with 4 sides that consists of two pairs of parallel opposite straight lines. Its angles measure $90^o$.

### What are the Characteristics of a Rectangle?

The characteristics of a rectangle are the following:

• It is considered a quadrilateral because it has four sides.
• It has four vertices, four edges and four angles.
• The pairs of sides are opposite, equal, and parallel.
• All four of it's angles measure $90^o$.

Do you think you will be able to solve it?

### What is a golden rectangle?

A golden rectangle is a rectangle where the proportion between its sides is the golden ratio (an irrational number also known as divine number). When dividing one of its sides by the other side, we obtain $\phi=1.61803\ldots$.

### How do you know if it is a golden rectangle?

To know if you are working with a golden rectangle, divide the larger side of the rectangle by the smaller side and see if that division gives you the result $\phi=1.61803\ldots$. If it does, it is a divine rectangle.

## examples with solutions for rectangle for ninth grade

### Exercise #1

True or false?

One of the angles in a rectangle may be an acute angle.

### Step-by-Step Solution

One of the properties of a rectangle is that all its angles are right angles.

Therefore, it is not possible for an angle to be acute, that is, less than 90 degrees.

False

### Exercise #2

Look at the rectangle ABCD below.

Side AB is 6 cm long and side BC is 4 cm long.

What is the area of the rectangle?

### Step-by-Step Solution

Remember that the formula for the area of a rectangle is width times height

We are given that the width of the rectangle is 6

and that the length of the rectangle is 4

Therefore we calculate:

6*4=24

24 cm²

### Exercise #3

True or false:

The sum of the angles of a rectangle is 360.

### Step-by-Step Solution

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

Since a rectangle is a type of quadrilateral, the sum of its angles is also equal to 360.

Likewise, we know that all the angles in a rectangle are right angles (90 degrees).

$90\times4=360$

True

### Exercise #4

Given the following rectangle:

Find the area of the rectangle.

### Step-by-Step Solution

We will use the formula to calculate the area of a rectangle: length times width

$9\times6=54$

54

### Exercise #5

Look at the parallelogram ABCD below.

What can be said about triangles ACD and ABD?

### Step-by-Step Solution

According to the side-angle-side theorem, the triangles are similar and coincide with each other:

AC = BD (Any pair of opposite sides of a parallelogram are equal)

Angle C is equal to angle B.

AB = CD (Any pair of opposite sides of the parallelogram are equal)

Therefore, all of the answers are correct.