Pythagorean Theorem - Examples, Exercises and Solutions

Look at the triangle in the diagram. How long is side AB?

To find side AB, we will need to use the Pythagorean theorem.

The Pythagorean theorem allows us to find the third side of a right triangle, if we have the other two sides.

You can read all about the theorem here.

Pythagorean theorem:

$A^2+B^2=C^2$

That is, one side squared plus the second side squared equals the third side squared.

We replace the existing data:

$3^2+2^2=AB^2$

$9+4=AB^2$

$13=AB^2$

We find the root:

$\sqrt{13}=AB$

$\sqrt{13}$ cm

Look at the triangle in the diagram. Calculate the length of side AC.

To solve the exercise, we have to use the Pythagorean theorem:

A²+B²=C²

We replace the data we have:

3²+4²=C²

9+16=C²

25=C²

5=C

5 cm

What is the length of the hypotenuse?

We use the Pythagorean theorem

$AC^2+AB^2=BC^2$

We replace the data we know:

$3^2+4^2=BC^2$

$9+16=BC^2$

$25=BC^2$

We extract the root:

$\sqrt{25}=BC$

$5=BC$

5

Look at the triangle in the diagram. How long is side BC?

To solve the exercise, it is necessary to know the Pythagorean Theorem:

A²+B²=C²

We replace the known data:

2²+B²=7²

4+B²=49

We input into the formula:

B²=49-4

B²=45

We find the root

B=√45

This is the solution. However, we can simplify the root a bit more.

First, let's break it down into prime numbers:

B=√(9*5)

We use the property of roots in multiplication:

B=√9*√5

B=3√5

This is the solution!

$3\sqrt{5}$ cm

Given the triangle ABC, find the length BC

To answer this question, we must know the Pythagorean Theorem

The theorem allows us to calculate the sides of a right triangle.

We identify the sides:

ab = a = 5
bc = b = ?

ac = c = 13

We replace the data in the exercise:

5²+?² = 13²

We swap the sections

?²=13²-5²

?²=169-25

?²=144

?=12

12 cm

What is the length of the side marked X?

We use the Pythagorean theorem:

$AB^2+BC^2=AC^2$

$15$

Look at the following triangle.

What is the value of X?

It is important to remember: the Pythagorean theorem is only valid for right-angled triangles.

This triangle does not have a right angle, and therefore, the missing side cannot be calculated in this way.

Cannot be solved

The triangle in the drawing is rectangular and isosceles.

Calculate the length of the legs of the triangle.

We use the Pythagorean theorem:

$AC^2+BC^2=AB^2$

Since the triangles are isosceles, the theorem can be written as:

$AC^2+AC^2=AB^2$

We replace the data we know:

$2AC^2=(8\sqrt{2})^2=64\times2$

We reduce the 2 and extract the root:

$AC=\sqrt{64}=8$

$BC=AC=8$

8 cm

Look at the triangle in the figure.

What is its perimeter?

To find the perimeter of a triangle, first we need to find all its sides.

Given two sides and only the perimeter remains to be found.

We can use the Pythagorean Theorem
$AB^2+BC^2=AC^2$
We replace all the known data:

$AC^2=7^2+3^2$
$AC^2=49+9=58$
We extract the square root:

$AC=\sqrt{58}$
Now that we have all the sides, we can add them up and thus find the perimeter:
$\sqrt{58}+7+3=\sqrt{58}+10$

$10+\sqrt{58}$ cm

Given the triangles in the drawing

What is the length of the side DB?

In this question, we will have to use the Pythagorean theorem twice.

A²+B²=C²

Let's start by finding side CB:

6²+CB²=(2√11)²

36+CB²=4*11

CB²=44-36

CB²=8

CB=√8

We will use the exact same way to find side DB:

2²+DB²=(√8)²

4+CB²=8

CB²=8-4

CB²=4

CB=√4=2

2 cm

Look at the triangle in the figure.

What is its perimeter?

To find the perimeter of a triangle, first we need to find all its sides.

Given two sides and only the perimeter remains to be found.

We can use the Pythagorean Theorem
$AB^2+BC^2=AC^2$
We replace all the known data:

$AC^2=7^2+3^2$
$AC^2=49+9=58$
We extract the square root:

$AC=\sqrt{58}$
Now that we have all the sides, we can add them up and thus find the perimeter:
$\sqrt{58}+7+3=\sqrt{58}+10$

$10+\sqrt{58}$ cm

The Egyptians decided to build another pyramid that looks like an isosceles triangle when viewed from the side.

Each side of the pyramid measures 150 m, while the base measures 120 m.

What is the height of the pyramid?

Since the height divides the base into two equal parts, each will be called X

Now we calculate X:$120:2=60$

Now we can calculate the height using the Pythagorean theorem:

$X^2+H^2=150^2$

We place the corresponding data:

$60^2+h^2=150^2$

We extract the root: $h=\sqrt{150^2-60^2}=\sqrt{22500-3600}=\sqrt{18900}$

$h=30\sqrt{21}$

$30\sqrt{21}$ m

The rectangle ABCD is shown below.

$BD=25,BC=7$

Calculate the area of the rectangle.

To find side DC we will use the Pythagorean theorem:

$(BC)^2+(DC)^2=(DB)^2$

Now we will replace the existing data in the theorem:

$7^2+(DC)^2=25^2$

$49+DC^2=625$

$DC^2=625-49=576$

We extract the root:

$DC=\sqrt{576}=24$

168

Look at the square below:

Is the sum of the two diagonals greater than the sum of the 3 sides of the square?

Let's look at triangle BCD, let's calculate the diagonal by the Pythagorean theorem:

$DC^2+BC^2=BD^2$

As we are given one side, we know that the other sides are equal to 4, so we will replace accordingly in the formula:

$4^2+4^2=BD^2$

$16+16=BD^2$

$32=BD^2$

We extract the root:$BD=AC=\sqrt{32}$

Now we calculate the sum of the diagonals:

$2\times\sqrt{32}=11.31$

Now we calculate the sum of the 3 sides of the square:

$4\times3=12$

And we reveal that the sum of the two diagonals is less than the sum of the 3 sides of the square.

11.31 < 12

No

ABCD is a parallelogram.

CE is its height.

CB = 5
AE = 7
EB = 2

What is the area of the parallelogram?

To find the area,

first, the height of the parallelogram must be found.

To conclude, let's take a look at triangle EBC.

Since we know it is a right triangle (since it is the height of the parallelogram)

the Pythagorean theorem can be used:

$a^2+b^2=c^2$

In this case: $EB^2+EC^2=BC^2$

We place the given information: $2^2+EC^2=5^2$

We isolate the variable:$EC^2=5^2+2^2$

We solve:$EC^2=25-4=21$

$EC=\sqrt{21}$

Now all that remains is to calculate the area.

It is important to remember that for this, the length of each side must be used.
That is, AE+EB=2+7=9

$\sqrt{21}\times9=41.24$

41.24