O Pythagorean theorem lists the measurements of the sides of a trianglerectangle in the following way:
On a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
Pythagoras' theorem is very important for Math, having influenced other great mathematical results. See also one of the proofs of the theorem and part of its creator's biography.
Also know: 4 most common mistakes in basic trigonometry
Pythagoras theorem formula
For application of Pythagorean theorem, it is necessary to understand the nomenclatures of the sides of a right triangle. O biggest side of the triangle is always opposite to the biggest angle, which is the 90° angle. This side is called hypotenuse and will be represented here by the letter The.
You other sides of the triangle are called peccaries and will be represented here by the letters B and ç.
Pythagoras' theorem states that the following relationship is valid:
Thus, we can say that the square of the measure of the hypotenuse is equal to the sum of the squares of the measures of the legs.
Proof of the Pythagorean Theorem
Let's see below one of the ways to show the veracity of Pythagorean theorem. For this, consider a square ABCD with measuring side (b + c), as shown in the figure:
O first step consists of determining the area of square ABCD.
THEA B C D = (b + c)2 = b2 + 2bc + c2
O second step consists of determining the area of the EFGH square.
THEE F G H = the2
We can see that there are four congruent triangles:
O third step is to calculate the area of these triangles:
THEtriangle = b·c
2
O fourth step and last requires calculating the area of square EFGH using the area of square ABCD. See that if we consider the area of square ABCD and withdraw the area of the triangles, which are the same, only the square EFGH remains, so:
THEEFGH = THEA B C D – 4 · Atriangle
Replacing the values found in first, second and third step, let's get:
The2 = b2 + 2bc + c2 – 4 · bc
2
The2 = b2 + 2bc + c2– 2bc
The2 = b2 + c2
Mind Map: Pythagoras Theorem
*To download the mind map in PDF, Click here!
Pythagorean triangle
Any right triangle is called a Pythagorean triangle if the size of your sides satisfies the Pythagorean theorem.
Examples:
The triangle above is Pythagorean because:
52 = 32 + 42
The triangle below is not Pythagorean. Look
262 ≠ 242 +72
Read too:Applications of Trigonometric Laws of a Triangle: Sine and Cosine
Pythagorean Theorem and irrational numbers
Pythagoras' theorem brought with it a new discovery. When constructing a right triangle in which the peccaries are equal to 1, mathematicians, at the time, faced a great challenge, because, when finding the value of hypotenuse, an unknown number appeared. Look:
Applying the Pythagorean theorem, We have to:
The number found by mathematicians of the time today is called irrational.
Read too: Relationship between sides and angles of a triangle
solved exercises
question 1. Determine the value of x in the triangle below.
Resolution:
Applying the Pythagorean theorem, we have the following:
132 = 122 + x2
solving the potencies and isolating the unknown x, we have:
x2 = 25
x =5
Question 2. Determine the measure ç of the legs of an isosceles right triangle in which the hypotenuse measures 30 cm.
Resolution:
We know that the isosceles triangle has two equal sides. Then:
Applying the Pythagorean theorem, we will have to:
202 = c2 + c2
2c2 = 400
ç2 = 200
Thus, the measures of the legs of the triangle measure, respectively:
*Mental Map by Luiz Paulo Silva
Graduated in Mathematics
by Robson Luiz
Maths teacher
Source: Brazil School - https://brasilescola.uol.com.br/matematica/teorema-pitagoras.htm
