What Does the Pythagorean Theorem Say?

The Pythagorean Theorem can be formulated as follows: in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

In the right triangle shown in the image below, we use the first letters of the alphabet to indicate its sides:

a a and b b are the legs.

cc is the hypotenuse.

Using these, we can express the Pythagorean theorem in an algebraic form as follows:

c2=a2+b2 c²=a²+b²

How to solve the Pythagorean theorem

We can express the Pythagorean Theorem in a geometric form in the following way, showing that the area of the square (c c ) (square of the hypotenuse) is the sum of the areas of the squares (a a ) and (b b ) (squares of the legs).

geometrical form of the Pythagorean Theorem

Practice Pythagorean Theorem

Exercise #1

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What is the length of the hypotenuse?

Video Solution

Step-by-Step Solution

We use the Pythagorean theorem

AC2+AB2=BC2 AC^2+AB^2=BC^2

We replace the data we know:

32+42=BC2 3^2+4^2=BC^2

9+16=BC2 9+16=BC^2

25=BC2 25=BC^2

We extract the root:

25=BC \sqrt{25}=BC

5=BC 5=BC

Answer

5

Exercise #2

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

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Video Solution

Step-by-Step Solution

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:

A2+B2=C2 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:

32+22=AB2 3^2+2^2=AB^2

9+4=AB2 9+4=AB^2

13=AB2 13=AB^2

We find the root:

13=AB \sqrt{13}=AB

Answer

13 \sqrt{13} cm

Exercise #3

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

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Video Solution

Step-by-Step Solution

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

Answer

5 cm

Exercise #4

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What is the length of the side marked X?

Video Solution

Step-by-Step Solution

We use the Pythagorean theorem:

AB2+BC2=AC2 AB^2+BC^2=AC^2

Answer

15 15

Exercise #5

Look at the following triangle.

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What is the value of X?

Video Solution

Step-by-Step Solution

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.

Answer

Cannot be solved

Exercise #1

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

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Video Solution

Step-by-Step Solution

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!

Answer

35 3\sqrt{5} cm

Exercise #2

Given the triangle ABC, find the length BC

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Video Solution

Step-by-Step Solution

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

Answer

12 cm

Exercise #3

The triangle in the drawing is rectangular and isosceles.

Calculate the length of the legs of the triangle.

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Video Solution

Step-by-Step Solution

We use the Pythagorean theorem:

AC2+BC2=AB2 AC^2+BC^2=AB^2

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

AC2+AC2=AB2 AC^2+AC^2=AB^2

We replace the data we know:

2AC2=(82)2=64×2 2AC^2=(8\sqrt{2})^2=64\times2

We reduce the 2 and extract the root:

AC=64=8 AC=\sqrt{64}=8

BC=AC=8 BC=AC=8

Answer

8 cm

Exercise #4

Look at the triangle in the figure.

What is its perimeter?

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Video Solution

Step-by-Step Solution

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
AB2+BC2=AC2 AB^2+BC^2=AC^2
We replace all the known data:

AC2=72+32 AC^2=7^2+3^2
AC2=49+9=58 AC^2=49+9=58
We extract the square root:

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

Answer

10+58 10+\sqrt{58} cm

Exercise #5

Given the triangles in the drawing

What is the length of the side DB?

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Video Solution

Step-by-Step Solution

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

Answer

2 cm

Exercise #1

Look at the triangle in the figure.

What is its perimeter?

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Video Solution

Step-by-Step Solution

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
AB2+BC2=AC2 AB^2+BC^2=AC^2
We replace all the known data:

AC2=72+32 AC^2=7^2+3^2
AC2=49+9=58 AC^2=49+9=58
We extract the square root:

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

Answer

10+58 10+\sqrt{58} cm

Exercise #2

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?

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Video Solution

Step-by-Step Solution

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

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

Now we can calculate the height using the Pythagorean theorem:

X2+H2=1502 X^2+H^2=150^2

We place the corresponding data:

602+h2=1502 60^2+h^2=150^2

We extract the root: h=1502602=225003600=18900 h=\sqrt{150^2-60^2}=\sqrt{22500-3600}=\sqrt{18900}

h=3021 h=30\sqrt{21}

Answer

3021 30\sqrt{21} m

Exercise #3

The rectangle ABCD is shown below.

BD=25,BC=7 BD=25,BC=7

Calculate the area of the rectangle.

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Video Solution

Step-by-Step Solution

To find side DC we will use the Pythagorean theorem:

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

Now we will replace the existing data in the theorem:

72+(DC)2=252 7^2+(DC)^2=25^2

49+DC2=625 49+DC^2=625

DC2=62549=576 DC^2=625-49=576

We extract the root:

DC=576=24 DC=\sqrt{576}=24

Answer

24

Exercise #4

ABCD is a parallelogram.

CE is its height.

CB = 5
AE = 7
EB = 2

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What is the area of the parallelogram?

Video Solution

Step-by-Step Solution

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:

a2+b2=c2 a^2+b^2=c^2

In this case: EB2+EC2=BC2 EB^2+EC^2=BC^2

We place the given information: 22+EC2=52 2^2+EC^2=5^2

We isolate the variable:EC2=52+22 EC^2=5^2+2^2

We solve:EC2=254=21 EC^2=25-4=21

EC=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

21×9=41.24 \sqrt{21}\times9=41.24

Answer

41.24

Exercise #5

Given the rhombus in the drawing:

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What is the area?

Video Solution

Step-by-Step Solution

Remember there are two options to calculate the area of a rhombus:

Diagonal by diagonal divided by 2.

Side by the height of the side.

In the question, we are only given half of the diagonal and the side, which means we cannot use any of the formulas.

We need to find more data. Let's find the second diagonal:

Remember that the diagonals of a rhombus are perpendicular to each other, which means they form a 90-degree angle.

Therefore, all the triangles in a rhombus are right-angled.

Now we can focus on the triangle where the side and the height are given, and we will calculate the third side by the Pythagorean theorem:

a2+b2=c2 a²+b²=c² Replace the data:

32+x2=52 3^2+x^2=5^2 9+x2=25 9+x^2=25 x2=259=16 x^2=25-9=16 x=16=4 x=\sqrt{16}=4 Now that we have found half of the second diagonal, we can calculate the area by diagonal by diagonal:

Since the diagonals in a rhombus are perpendicular and cross each other, they are equal. Therefore our diagonals are equal:

3+3=6 3+3=6 4+4=8 4+4=8 Therefore, the area of the rhombus is:

6×82=482=24 \frac{6\times8}{2}=\frac{48}{2}=24

Answer

24