The Pythagorean Theorem can be formulated as follows: in a right triangle, thesquareof 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:

$a$ and$b$ are the legs.

$c$ is the hypotenuse.

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

$c²=a²+b²$

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

The Pythagorean Theorem is one of the most famous theorems in mathematics and one of the most feared topics among students. It is no coincidence that it is among the most common mathematical theorems and one that is likely to be encountered outside of your studies and exams.

This theorem is attributed to Pythagoras of Samos. Born in 570 BC, he was a Greek scholar to whom we also owe the word 'philosopher'.

The Pythagorean Theorem establishes the relationship between the three sides of a right triangle.

In this article, we will explain in a simple and practical way what the Pythagorean Theorem is and give you some examples. Let's jump in!

To begin with, it is essential to clarify some important points:

In every right triangle, thesides adjacent to the right angleare called "legs". The legs are the sides that form the right angle.

The longest side of a right triangle—the one opposite the right angle—is called the "hypotenuse".

What is a Theorem?

We can use the Pythagorean Theorem to understand what a theorem is.

A theorem is a demonstrable statement that links two propositions. We start from a first proposition that we call a hypothesis to assert a second proposition that we call a thesis.

The statement of a theorem affirms that if the hypothesis is true, then the thesis is also true.

The proof of a theorem is the most difficult part and is usually left to mathematicians. The important thing is that once a theorem is proved, we can confidently use the statement of the theorem as a permanent truth.

Returning to the Pythagorean Theorem, let us highlight which is the hypothesis and which is the thesis. To do this, we reformulate the statement of the theorem using the expressions if and then as follows:

The Pythagorean Theorem states that:

If:

A triangle is right angled (a triangle containing an angle of $90^o$ ) (hypothesis).

Then:

The square of the longest side of the triangle is the sum of the squares of the other two sides (thesis).

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Question 1

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

In the Pythagorean Theorem, the reciprocal of the theorem is also true, For example:

If:

The square of the longest side of a triangle is the sum of the squares of the other two sides (hypothesis).

Then:

2. The triangle is right angled (the value of one of the angles of the triangle is $90^o$ ) (thesis).

What is the Pythagorean Theorem Used For?

The Pythagorean Theorem is arguably a cornerstone of Cartesian geometry and has therefore become an important driving force in the development of the sciences as we know them today.

The importance of this theorem comes from the significance of the right triangle, which is a triangle that links a horizontal line with a vertical line (the legs of the triangle). The horizontal and vertical lines always form an angle of $90^o$.

The Pythagorean Theorem is applicable in all areas of science as they all share a mathematical basis.

If you are interested in learning more about other triangle topics, you can have a look at one of the following articles:

Calculate the length of $X$ in the following right triangle:

The image shows a right triangle with two known side lengths and one unknown side length. We want to work out the length of the third side—in this case, one of the legs.

The Pythagorean theorem states that the following is true in a right triangle:

$c²=a²+b²$

In our right triangle:

$a= 3$

$b=X$

$c=5$

By substituting the values of our triangle in to the algebraic expression of the Pythagorean Theorem, we obtain the following equation:

$5²=3²+X²$

$25=9+X²$

To work out the value of $X$, we must begin by subtracting $9$ from each side of the equation.

$25-9=X²$

$16=X²$

Next, by removing the square root from both sides of the equation, we get the value of $X$.

$\sqrt{16}=X$

$X=4$

Therefore, the answer is: $4$.

Exercise 3

What is the value of $X$ in the triangle shown in the following image?

Solution:

We are dealing with a right triangle which is also isosceles since two of its sides have the same length.

In this case, we know the length of the hypotenuse of the right triangle and we want to work out the length of each leg, knowing that both legs have the same value.

The Pythagorean theorem says that the following is true for a right triangle:

$c²=a²+b²$

In our right triangle:

$a= X$

$b=X$

$c=10$

By substituting the values of our triangle in to the algebraic expression of the Pythagorean Theorem, we get the following equation:

$10²=X²+X²$

$100=2X²$

To work out X, we must begin by dividing each side of the equation by $2$.

$\frac{100}{2}=\frac{2X²}{2}$

$50=X²$

Finally, we need to remove the square root from both sides of the equation to obtain the value of $X$.

What is the Most Common Use of the Pythagorean Theorem?

The Pythagorean Theorem is mainly used in exercises related to right triangles in which the length of both legs is given to find the length of the hypotenuse.

The Inverse of the Pythagorean Theorem

There is also the inverse theorem by which we can prove that a given triangle is right-angled: atriangle in which the sum of both legs squared is equal to the hypotenuse squared is a right-angled triangle.