Pythagoras Theorem: A Comprehensive Guide

Every maths student will be familiar with the term Pythagoras Theorem, also known as Pythagorean Theorem, and even students who are not majoring in mathematics are likely to have studied it in school.

Given that the triangle must have a right angle, this theorem connects the three sides of a triangle in Euclidean geometry. It is possible to allude to the qualities and different dimensions of the theorem geometrically, and it is commendable that it has been applied to real-world situations.

Learning about Pythagoras Theorem and Its Use might be impossible, along with its geometrical and metric representations, in a single post. Since doing so would need in-depth knowledge of angles and their uses.

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Introduction: Learn the Secrets of Triangles and Mathematics

The Greek mathematician Pythagoras of Samos developed the Pythagoras theorem. He was an Ionian Greek philosopher from antiquity. He founded a community of mathematicians who practiced rigorous number theory and had monastic lifestyles.

The Greek mathematician Pythagoras finally proved the theorem, which is why it is known by that name. Although it was developed over a long period of time, it is now essential to use it in order to cope with maths practical circumstances.

What Is Pythagoras Theorem?

Do you know that Pythagoras' Theorem is referred to as one of the fundamental principles in geometry, known by the name of the ancient Greek mathematician Pythagoras?

It basically related to a right-angle triangle, which means that the square of the length of the longest side opposite the right angle is equal to the sum of the squares of remaining the two sides of the triangle.

Mathematically, it can be written as “a² + b² = c²”, where 'a' and 'b' are the lengths of the two shorter sides, and 'c' is the length of the hypotenuse.

This theorem can be utilized in diverse fields, starting from architecture to engineering, and plays a vital role in understanding spatial relationships and solving real-world problems involving right triangles.

What is the formula for Pythagoras' theorem?

The formula for Pythagoras' theorem is a² + b² = c². In this equation, c is the longest side of a right-angled triangle. This line is also known as the hypotenuse.

Pythagorean Theorem Examples & Solutions

Question 1: Identify the hypotenuse of a triangle whose lengths of two sides are 4 cm and 10 cm.

Solution: Using the Pythagoras theorem,

Hence, the hypotenuse of the triangle is 10.77 cm.

An Overview of a Right-Angled Triangle

In trigonometry, a right-angled triangle is a triangle in which one of the angles has the measurement value of theta, which is frequently used to represent the angle with a measure of 90.

The two sides that make up a right angle are known as the opposite side and the adjacent side, respectively. The hypotenuse side, or simply the hypot, is the side that forms the right angle.

Pythagorean Triangle and Triples

Consider a right-angled triangle with the sides a, b, and c that trifurcate into three halves. The adjacent and opposing sides are referred to as b and c, respectively, while the hypotenuse side is referred to as a. In every triangle, the hypotenuse side is the longest and takes up a significant amount of space.

The term "Pythagorean triangle" and the lengths of sides "a," "b," and "c" jointly refer to a right-angled triangle whose three sides are all integers.

Statement Of Pythagoras Theorem

In a right-angled triangle, the hypotenuse's square equals the sum of the squares of the other two sides.

Now, imagine a cake-cutting ceremony between a father, mother, and their daughter. The daughter sliced the cake slantwise, creating four sides labeled A, B, C, and D.

She made diagonal cuts on endpoints A and C, splitting the cake into two triangles. Her mother instructed her to take the first triangle.

The bigger of the three pieces, which resemble a right-angled triangle, is the hypotenuse side, which the parents handed to the birthday girl.

The other two pieces, which are opposite and adjacent, are divided between the parents. The second half of the triangular cake may be divided into three halves using the same trifurcation.

The foremost principle of the Pythagoras theorem is that if we measure the sum and squares of the sides b and c, they should be equal to the sum and square of the child's portion, which is a.

It may be expressed mathematically as follows.

= +

a – the largest portion, child’s share, hypotenuse side

b – Father’s share, Opposite side

c – Mother’s share, Adjacent side

Applications of Pythagoras Theorem

We would be more interested in knowing the Pythagoras theorem applications that we decapitate in day-to-day living circumstances, even if it is important to comprehend the fundamental ideas such as the theorem statement and its mathematical representation.

Construction and engineering disciplines

Most architects employ the Pythagoras theorem method to determine the value, and it is fairly simple to determine the diameter of a certain sector, given length or width is known. In engineering applications, it is mostly employed in two dimensions.

Face detection in surveillance cameras

Face recognition is a technology that is increasingly widely used nowadays and helps security departments investigate crimes more quickly. Using the Pythagorean theorem concept, the security camera's lens precisely projects the distance between the spot where the person is seen and the surveillance camera.

Interior design and furniture making

According to the basic idea, the Pythagoras idea may be used very simply if the square cardboard can be readily transformed into a triangle by cutting diagonally. Most woodworks are created using a system that helps the designers work more quickly.

Navigation

This approach is used by seafarers to determine the quickest route and shortest distance to their destinations, which is an incredible truth.

Surveying

Surveyors typically use this method to locate the steep, hilly terrain because once they know the region's horizontal extent, it will be simpler for them to apply the Pythagorean principle to the remaining terrain. The surveyor who creates the trail can use a telescope to compare the constant distance to the variable one.

Physics:

The Pythagoras Theorem is applicable in physics, particularly in problems related to vectors and forces. It helps determine the net force acting on an object when forces are applied at different angles.

Computer Graphics:

In computer graphics and 3D modeling, the theorem is used to calculate distances, angles, and dimensions, ensuring that virtual objects are accurately represented.

Medical Imaging:

Pythagoras Theorem is applied in medical imaging techniques like ultrasound and MRI to measure distances within the human body, aiding in diagnosis and treatment planning.

Though the Pythagoras theorem has many applications, only a small number are covered in this article. Many theorems are solely expressed in terms of the theorem's basic idea. I believe after reading this blog, you will have a good understanding of what the Pythagoras theorem is all about and what is it uses and applications are in our daily lives.

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• Firstly, Pythagoras Theorem frequently appears in assignments of students who are pursuing their degree in mathematics and that involve calculating distances, areas, and angles within right-angled triangles. Students are tasked with applying the theorem to find missing sides or angles, making it a common subject of assignments in math courses.

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