# Rectangle for ninth grade - Examples, Exercises and Solutions

## 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.

## 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.

## examples with solutions for rectangle for ninth grade

### Exercise #1

Look at the rectangle below.

Side AB is 4.8 cm long and side AD has a length of 12 cm.

What is the perimeter of the rectangle?

### Step-by-Step Solution

In the drawing, we have a rectangle, although it is not placed in its standard form and is slightly rotated,
but this does not affect that it is a rectangle, and it still has all the properties of a rectangle.

The perimeter of a rectangle is the sum of all its sides, that is, to find the perimeter of the rectangle we will have to add the lengths of all the sides.
We also know that in a rectangle the opposite sides are equal.
Therefore, we can use the existing sides to complete the missing lengths.

4.8+4.8+12+12 =
33.6 cm

33.6 cm

### Exercise #2

The width of a rectangle is equal to $4^2$ cm and its length is $5$ cm.

Calculate the area of the rectangle.

### Step-by-Step Solution

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

According to the existing data:

$4^2\times5^=S$

$S=16\times5$

$S=80$

80

### Exercise #3

The area of the rectangle below is equal to 63.

AC = 7

How long is side AB?

### Step-by-Step Solution

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

$AB\times AC=S$

We place the existing data into the formula:

$AB\times7=63$

$7AB=63$

We divide both sides by 7:

$AB=9$

9

### Exercise #4

The area of a rectangle is equal to 8.

Calculate the perimeter of the rectangle.

### Step-by-Step Solution

According to the properties of the rectangle, all pairs of opposite sides are equal.

$AB=CD=8$

$AC=BD=2$

Now we calculate the perimeter of the rectangle by adding all the sides:

$4+4+2+2=8+4=12$

12

### Exercise #5

The area of the rectangle below is equal to 24.

Calculate the perimeter of the rectangle.

### Step-by-Step Solution

Given that in a rectangle all pairs of opposite sides are equal to each other, it can be argued that:

$AB=CD=6$

$AC=BD=4$

Now calculate the perimeter of the rectangle by adding all the sides:

$4+4+6+6=$

$8+12=20$

In other words, the data of the rectangle's area is unnecessary, since we already have all the data to calculate the perimeter, and we do not need to calculate the other sides.

20

## examples with solutions for rectangle for ninth grade

### Exercise #1

What is the perimeter of the white area according to the data?

26

### Exercise #2

Below is the rectangle ABCD.

It has an area of 42 cm² and side AD is equal to 12 cm.

What is the length of side DC?

### Step-by-Step Solution

Remember that to calculate the area of the rectangle, we multiply the length by the width.

Therefore:

$42=12\times CD$

$42=12CD$

Divide both sides by 12:

$3.5=CD$

3.5

### Exercise #3

The perimeter of the rectangle below is equal to 30.

What is the area of the rectangle?

### Step-by-Step Solution

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

$AC\times AB=S$

We replace the existing data:

$5\times10=50$

That is, the information that the perimeter of the rectangle is equal to 30 is unnecessary, since all the data to calculate the area already exist and it is not necessary to calculate the other sides.

50

### Exercise #4

Look at the given rectangle made of two squares below:

What is its area?

### Step-by-Step Solution

In a square all sides are equal, therefore we know that:

$AB=BC=CD=DE=EF=FA=5$

The area of the rectangle can be found in two ways:

1. Find one of the sides (for example AC)

$AC=AB+BC$

$AC=5+5=10$

and multiply it by one of the adjacent sides to it (CD/FA, which we already verified is equal to 5)

$5\times10=50$

2. Find the area of the two squares and add them.

The area of square BCDE is equal to the multiplication of two adjacent sides, both equal to 5.

$5\times5=25$

Square BCDE is equal to square ABFE, because their sides are equal and they are congruent.

Therefore, the sum of the two squares is equal to:

$25+25=50$

50

### Exercise #5

Look at the two rectangles in the figure:

What is the area of the white area?

### Step-by-Step Solution

As we know that EFGD is a rectangle, we also know that DE is equal to 2 and DG is equal to 4

In a rectangle, each pair of opposite sides are equal and parallel, therefore:

$ED=FG=2$

$DG=EF=4$

Now we calculate the area of the orange rectangle EFGD by multiplying the length by the width:

$2\times4=8$

Now we calculate the total area of the white rectangle ABCD:

$AD=AE+ED=2+2=4$

$DC=DG+GC=4+5=9$

The area of the entire rectangle ABCD is:

$4\times9=36$

Now to find the area of the white part that is not covered by the area of the orange rectangle, we will subtract the area of the rectangle EFGD from the rectangle ABCD:

$36-8=28$