The Pythagorean Theorem

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:

• A triangle is a polygon with three sides.
• The angles of any triangle add up to $180^o$ degrees.
• A right triangle is a triangle in which one of the angles equals $90^o$ degrees.
• A right angle is an angle of $90^o$ degrees.
• In every right triangle, the sides adjacent to the right angle are 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".

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).

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

If:

1. 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).

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:

• Acute triangle
• Obtuse triangle
• Scalene Triangle
• Equilateral triangle
• Isosceles Triangle
• The edges of a triangle
• Area of a right triangle
• Height of a triangle
• How to calculate the perimeter of a triangle?
• Congruence of right triangles (in the context of the Pythagorean Theorem)
• Applying the Pythagorean Theorem to an orthohedron or cuboid

Now that we have laid the groundwork, we can continue with the exercises!

Below we present several problems with the Pythagorean theorem:

Find the value of $X$ in the following triangle:

Solution:

The picture shows a triangle. We know the length of two of its sides, but we want to work out the value of the third side.

We also know that the triangle shown is a right triangle because a small box indicates a right angle.

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

$c²=a²+b²$

In our right triangle:

$a= 3$

$b=4$

$c=X$

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

$X²=3²+4²$

$X²=9+16$

$X²=25$

If we now remove the square root from both sides of the equation, we can work out $X$ and obtain the value we are looking for:

$X=\sqrt{25}$

$X=5$

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: $4$.

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 $X$.

$\sqrt{50}=X$

$X=7.07$

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.

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

Given the triangle:

Task:

What is the value of $X$?

Solution:

Don't fall into the trap!

Remember, in the Pythagorean Theorem only a right triangle can be used. This triangle is an obtuse triangle, so the theorem does not apply to it.

Answer:

It is not possible to calculate the answer using the Pythagorean theorem.

Assignment:

Given the triangle $\triangle ABC$, find the length of $BC$.

Solution:

Apply the formula.

Given the triangle $\triangle ABC$ in the drawing, write the Pythagorean Theorem for the right triangle: $\triangle ABC$

$AB²+BC²=AC²$

We substitute in the known lengths:

$5²+BC²=13²$

$25+BC²=169$

$BC²=169-25=144$,$\sqrt{}$

$BC=12$

Answer:

$12$ cm.

Task:

Given the triangles in the drawing, what is the length of the side $DB$?

Solution:

We start with the triangle $\triangle ABC$, writing down the Pythagorean Theorem as an equation:

$BC²+BA²=AC²$

We then substitute in the given sides from the diagram:

$BC²+6²=(2\sqrt{11})²$

$BC²+36=2²\sqrt{11}²=4\times11=44$

$BC²+36=44$ /$-11$

$BC²=44-36=8$ /$\sqrt{}$

$BC=\sqrt{8}$

Now, we write down the equation for the triangle $\triangle BCD$:

$DC²+DB²=BC²$

Finally, we insert the values of the sides $CD$, $BC$:

$2²+DB²=(\sqrt{8})²$

$4+DB²=8$ /$-4$

$DB²=4$ / $\sqrt{}$

$DB=2$

Answer:

$DB=2$

Task:

The triangle $\triangle ABC$ is given.

The ratio of $BC$ to the hypotenuse $AC$ is $1:4$:

$AB=3\sqrt{15}$

Solution:

Write the Pythagorean Theorem as an equation for the triangle: $ABC$:

$AB²+BC²=AC²$

For the hypotenuse $AC$ ratio with $BC$ we can write:

$\frac{BC}{AC}=\frac{1}{4}$

Therefore:

$BC=\frac{1}{4}AC$

Substituting our values in to the formula:

$(3\sqrt{15})²+(\frac{1}{4})²AC²=AC²$

$3²\sqrt{15}²+\frac{1}{16})²AC²=AC²$

We perform the operations in the formula:

$-\frac{1}{16}AC²$

$9\times15=AC²-\frac{1}{6}AC²=\frac{15}{16}AC²$ / We multiply by $\frac{16}{15}$

$9\times15\times\frac{16}{15}=AC²$ / We apply the root.

$\sqrt{}$

$\sqrt{9\times16}=\sqrt{9}\times\sqrt{16}=3\times4=12=AC$

Answer:

$AC=12cm$

Yoel goes up a bicycle ramp to a maximum height of $3$ meters above the ground.

All other measurements are shown in the figure:

On the way up Yoel, travels at a speed of $2$ meters per second.

On the way down, he travels at a spreed of $6$ meters per second.

Task:

How long will it take Yoel to go over the ramp?

Solution:

Ascent:

Find the hypotenuse of the right triangle that has legs measuring $3$ and $7$ meters long.

$3²+7²=(hypotenuse)²$

$9+49=(hypotenuse)²$

$hypotenuse=\sqrt{58}=7.62$

$time=\frac{distance}{speed}=\frac{7.62}{2}=3.81\text{ }$

Descent:

Find the hypotenuse of the right triangle the legs of which are $3$ and $4$ meters long.

$(hypotenuse)²=3²+4²$

$\sqrt{}$ / $(hypotenuse)²=9+16$

$5=\sqrt{25}$

Hypotenuse: $5$ meters.

$\text{time}=\frac{distance}{speed}=\frac{5}{6}=0.83$

$\text{time}=3.81+0.83=4.64$

Answer:

Yoel will cross the ramp in $4.64$ seconds.