▷ Pythagorean Theorem 【Definition and applications】

Before we start talking about Pythagoras theoremand in order to understand what it is about, we must remember two things:

What are right angles?

A right angle is one that measures 90º.

Let’s see:

In it right angle 1 We see that the measurement is expressed directly and that it is 90º.

Generally, right angles are marked by completing a small square, as seen in the right angle 2. In these cases the measurement is even omitted because it is understood that it is 90º.

What is a right triangle? What are its sides called?

A right triangle is one that has an angle of 90º, also called right angle.

Let’s look at some examples of right triangles:

The triangle that we present in Example 1 is the most classic. It is the one that appears most frequently in books.

But it is very important that you see well how the others are also right triangles because they have a right angle, regardless of the position in which the triangle or the right angle is located!

The sides of the right triangle are called legs and hypotenuse.

Let’s look at the previous triangles with their sides identified:

Since you know the main elements that make it up, Let’s see what the Pythagorean theorem is about:

First let’s take a right triangle with measurements 3, 4 and 5.

From what we have already seen, 3 and 4 correspond to the measurements of the legswhich are the shortest sides, and 5 corresponds to the measurement of the hypotenusewhich is the longest side.

Now let’s draw a square on each of its sides.

Let’s calculate the area corresponding to each square.

Remember that the area of ​​a square is calculated by multiplying the side measurement by itself, or by squaring the side measurement, which is the same thing.

Area of ​​square = l2being l the side measurement.

We are then left with the following:

He area of the square of longest leg is 16

He area of the square of shorter leg is 9

He area of the square of the hypotenuse is 25

Notice that The area of ​​the square of the hypotenuse is equal to the sum of the squares of the legs.

What we have just stated is the Pythagoras theorem!

Expressed formally, and generalized for any triangle, it would be:

If we call ayba los catos of the right triangle, and we call ca the hypotenusethe following equality holds:

From here the following formulas are derived:

Let’s look at other examples of the theorem.

Approach 1

Let triangle ABC be a right triangle with the following measurements:

How long is side AC?

Solution

We identify the conditions of the approach

As you can see, the triangle in the previous figure is right angled.

So sides AB and BC are its legs and side AC is the hypotenuse.

Side AB measures 5 cm and side BC measures 12 cm.

They are asking us to find the length of the hypotenuse, which is side AC.

We apply the Pythagorean theorem

We substitute the conditions of the approach in the Pythagorean theorem.

We resolve to get what they ask of us, which is the measure of A.C.

Therefore, The measurement of side AC is 13 centimeters.

Approach 2

Look at the following triangle and calculate the missing measurement:

Solution

We identify the conditions of the approach

In this case we have the right triangle PQR.

The cathet PQ measures 15, the hypotenuse QR measures 17.

The measurement of the leg is missing PRwhich is what they ask of us.

We apply the Pythagorean theorem, in this case we will calculate a leg, and we solve.

We substitute the conditions of the approach in the Pythagorean theorem.

Where from, the measurement of the PR leg is equal to 8.

A little history of the Pythagorean theorem

The origin of what is known today as the Pythagorean theorem dates back to Ancient Egypt. There are indications that this civilization knew and used on a practical level the pythagorean triplets.

A three-thousand-year-old Egyptian mural shows how members of the Egyptian troops carried a rope with 12 equidistant knots. When this string is put into the shape of a triangle with sides of length 3, 4, 5 units, the triangle is right angled.

It is said that the Egyptians used this triangle practically.

Ancient Egypt was a civilization that formed along the Nile River. Egypt was very economically prosperous and this was largely due to the periodic flooding caused by the Nile in its Delta.

When the waters of the Nile returned to their channel, the lands were ready to begin a great harvest. The only bad thing was that when the land flooded, the boundaries of the plots were lost. This led to many measurements of the land being made, because year after year the plots had to be demarcated.

The question was: How to draw perpendicular lines to demarcate plots that were generally rectangular?

This doubt was reduced to the need to construct perpendicular lines that gave rise to the rectangles.

What did the Egyptians do to solve this problem?

It was here where it emerged the rope with 12 knots all at the same distance. The Egyptians stuck the end of the rope into the ground and built a triangle in which its sides had 3, 4 and 5 knots.

The first thing they did was tie 12 knots on a rope, all at equal distances.

Then with that rope they made a triangle with sides of 3, 4, 5 units long. This triple determined a right angle inside the triangle. In this way they were able to assemble the rectangles to demarcate the plots.

Those in charge of rearranging the plots did their work with the support of this sacred triangle. Notice how the rectangle that the Egyptians needed so much is formed.

It would later be Pythagoras, who on one of his trips to Egypt discovered this metric property used by the Egyptians, who would carry out the formal demonstration. Thus giving way to what is known today as the Pythagorean Theorem.

Geometric verification of the Pythagorean theorem

To visualize the well-known Pythagorean theorem, we will work with the idea of ​​puzzles and approach it just as Plato did, who proved this theorem for an isosceles and right triangle.

The first thing we will do is construct a triangle that is isosceles and right angled:

Now we will draw a square on each of the legs. We will also construct a square on the hypotenuse.

Now we will draw the two diagonals of each of the squares that have been built on the legs.

We see that the surface of each of the squares drawn on the legs is covered by four equal triangles.

Now we will move, as if it were a puzzle, the 8 triangles of the legs towards the square drawn on the hypotenuse, in such a way that this entire surface is covered.

It can be seen that the area of ​​the surfaces built on the legs is equal to the area of ​​the surface built on the hypotenuse. This relationship is known as the tPythagorean theorem.

In every right triangle the sum of the areas of the squares drawn on the legs is equal to the area of ​​the square drawn on the hypotenuse.

Another verification of the Pythagorean theorem.

This simple puzzle proves the Pythagorean theorem in the case that the length of one of the legs is twice that of the other.

By rearranging the pieces, you can see that the sum of the areas of the squares built on the legs is equal to the area of ​​the square drawn on the hypotenuse.

Extending the Pythagorean theorem

We have seen that the Pythagorean theorem holds for squares that are built on each of the sides of a right triangle. Is it true that the Pythagorean theorem holds for any similar figures? That is, for figures that have the same shape, but not necessarily the same size.

To visualize this new idea about the Pythagorean theorem we will draw a right triangle whose sides measure 3 cm, 4 cm and 5 cm.

Now we will draw three equilateral triangles, for this we will consider the lengths 3 cm, 4cm and 5cm of the triangle ABC.

Now we will cut out the equilateral triangles whose sides correspond to the legs and hypotenuse of triangle ABC.

Cut the equilateral triangle with a side of 3 cm into three pieces.

We proceed to completely cover the area of ​​the triangle with sides equal to 5 cm with the pieces obtained from the 4 cm and 3 cm triangles.

It is observed that the sum of the areas of the triangles drawn on the legs of triangle ABC is equal to the area of ​​the hypotenuse of this same triangle.

In this way it can be seen that the Pythagorean theorem holds for any similar figures.

Next we leave you other cases of similar figures for which the Pythagorean theorem also holds.

Some applied problems of the Pythagorean theorem

Problem 1: The lighthouse and the boat

From the highest part of a 45 m high lighthouse you can see a boat 53 m away. Find the distance from the foot of the lighthouse to the boat.

Solution

By graphically representing the situation we realize that the required distance corresponds to one of the legs of the right triangle that is formed with the foot of the lighthouse (A), the tip of the lighthouse (B) and the boat (C).

The distance from the foot of the lighthouse to the boat corresponds to the AC side.

Now we apply the theorem to find the measure of the leg:

The distance from the foot of the lighthouse to the boat is 28 meters.

Problem 2: Checking if the triangle is right angled

We have the following triangle:

Is triangle JKL a right triangle?

To answer this we will rely on the converse of the Pythagorean theorem what does it say:

If in any triangle the square of its longest side is equal to the sum of the squares of the other two sides, then that triangle is right angled.

In this case it must be fulfilled that:

JL² = JK² + KL²

Let’s see:

80² = 64² + 48²

6400 = 4096 + 2304

6400 = 6400

Since the equality of the reciprocal of the Pythagorean theorem is fulfilled, then we can say that the triangle JKL is right angled.

Finally, we leave you some activities to practice what you have learned:

Select a game 👇 👇 👇

The right triangle

Origin of the Pythagorean theorem

Statement of the Pythagorean theorem

Proof of the Pythagorean theorem

Formulas derived from the Pythagorean theorem

Application of the Pythagorean theorem in problems

Questionnaire

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