The Pythagorean theorem can be considered as the most important theorem in geometry. This theorem allows us to use an algebraic equation to solve geometric problems. According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the two legs. Recall that the hypotenuse is the side opposite the right angle in a right triangle and the legs are the other two sides of the triangle.
Here, we will look at a little history of this theorem. In addition, we will learn how to demonstrate it and use it to solve some practice examples.
History of the Pythagorean theorem
Pythagoras of Samos (569-500 BC) was born on the island of Samos in Greece and traveled extensively in Egypt, learning mathematics and other things. More details about how his first years were are unknown. Pythagoras began to be recognized and to form status by founding a group known as the Brotherhood of Pythagoras, whose objective was the study of mathematics.
The brotherhood of Pythagoras had various aspects of a cult such as symbols, rituals, and prayers. Furthermore, Pythagoras believed that “number governs the universe,” and members of the Pythagorean group gave numerical values to many objects and ideas. These numerical values, in turn, were endowed with mystical and spiritual qualities.
A legend has it that when Pythagoras finished his famous theorem, he sacrificed 100 oxen. Although this theorem is attributed to Pythagoras, it is not possible to know with certainty whether he was truly the real author.
The Brotherhood of Pythagoras worked on many geometric tests, but it is difficult to know who tested what, as the group always tried to keep their findings secret.
Unfortunately, this vow of secrecy prevented an important mathematical idea from being publicly known. The Brotherhood of Pythagoras had discovered irrational numbers. When we consider an isosceles right triangle with legs of measure 1, the hypotenuse will measure the square root of 2.
However, we know that this number cannot be expressed as a length that can be measured with fractional parts, and that deeply disturbed the Pythagoreans, who believed that “Everything is number.”
Pythagorean theorem formula
The Pythagorean theorem indicates that, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. For example, in the right triangle below, the hypotenuse is side c and the legs are sides a and b.
Therefore, by the Pythagorean theorem, we have:
This formula allows us to find the length of the hypotenuse if we know the length of the two legs. Alternatively, we can use the formula to find the length of one of the legs if we know the length of the hypotenuse and the other leg.
Proofs of the Pythagorean theorem
There are several methods that can be used to prove the Pythagorean theorem. However, the most common are the Pythagorean proof and the proof through algebra. If you are interested in additional proofs, check out this article.
We can start with the following right triangle:
Lengths a and b represent the legs and length c represents the hypotenuse. Using four of these triangles, we are going to form a big square with sides of length :
The sides of the inner square have a length of c since they are equal to the hypotenuses of the triangles. This means that its area is equal to .
We are going to rearrange the triangles as follows:
This forms two squares with the areas and .
The area of both large squares formed is the same in both cases. Since the triangles used are the same, the area of the squares and is equal to the area of the square . That is, we have:
Proof using algebra
We can use a diagram similar to the one we used earlier. Four right triangles are used to form a large square and an inner square with sides equal to the hypotenuses of the triangles.
The lengths a and b are the legs of the triangles and c is the hypotenuse. The large square formed has sides of length . This means that its area is equal to
Since the inner square has sides of c, its area is equal to . Additionally, we can see that the area of the large square is equal to the area of the four triangles plus the area of the inner square. This means that we have:
Examples of application of the Pythagorean theorem
The following are some examples of how to solve problems with the Pythagorean theorem.
If we have the following right triangle, what is the value of X?
Solution: In this triangle, 3 and 4 represent the legs and X represents the hypotenuse. Therefore, using the Pythagorean theorem, we have:
The value of X is equal to 5.
Find the value of Y in the right triangle below.
Solution: In this case, we have to find the value of one of the legs of the triangle. Therefore, we have to use the Pythagorean theorem and solve for a or for b:
The value of Y is 5.
A right triangle has a hypotenuse of 13 m and a leg of 9 m. What is the length of the other leg?
Solution: We can recognize the values c=13 and a=9. Therefore, we simply use the Pythagorean theorem with these values and solve for b which corresponds to the value of the other leg:
The length of the other leg is equal to 9.38 m.
If the square of the hypotenuse of an isosceles right triangle is equal to 128 m². What is the length of one of the legs?
Solution: Recall that an isosceles right triangle is a triangle that has a right angle and that has two legs with the same length. The following is a diagram of this triangle:
Therefore, applying the Pythagorean theorem with this triangle, we have the following:
The length of one of the legs is equal to 8 m.
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