# Pythagorean Theorem Formula

## Pythagorean Theorem Formula

The Pythagorean theorem, sometimes known as Pythagoras’ theorem, is a fundamental relationship in Euclidean geometry between the three sides of a right triangle. It states that the area of the square with the hypotenuse (the side opposite the right angle) equals the sum of the areas of the squares with the other two sides. This theorem can be expressed as an equation connecting the lengths of the sides a, b, and hypotenuse c, which is commonly known as the Pythagorean Theorem Formula. Students need to pay attention to the proof of Pythagoras theorem. It is necessary to revise it as well. Solving questions from time to time helps students to get a thorough understanding of the topics. It is crucial to keep assessing the exam preparation in order to bring necessary changes in the process of learning. All the difficult topics need to be focused on more. Students having concerns in practising questions that are related to difficult topics can take help from the NCERT solutions available on Extramarks.  Students having difficulties in preparing for their upcoming examinations must take help of Extramarks. They can score higher marks in examinations by utilising study materials of Extramarks. There are numerous practice materials available that can help students enhance their understanding in each topic.

What is the Pythagoras Theorem?

The relationship between the three sides of a right-angled triangle is explained by the Pythagoras theorem, also known as the Pythagorean Theorem Formula. The Pythagorean Theorem Formula states that the square of a triangle’s hypotenuse is equal to the sum of its other two sides’ squares.

### History of Pythagoras Theorem

The theorem is named after the Greek philosopher Pythagoras, who lived about the year 570 BC. The theorem has been demonstrated multiple times using various methods, maybe more than any other mathematical theorem. The proofs are varied, including geometric and algebraic proofs, and some date back thousands of years. The Pythagorean Theorem Formula, also known as Pythagoras’ theorem, describes the relationship between the sides of a right triangle. A right triangle is one in which one of the three angles is 90 degrees. Legs and hypotenuse are the sides of a right triangle. Pythagoras or his disciples built the first known algebraic proof of the theorem, and famous writers such as Plutarch and Ciceron credited him with discovering it. As a result, he is credited with creating this lovely connection between the sides of a right triangle.

In analytic geometry, Euclidean distance satisfies the Pythagorean relation when Euclidean space is represented by a Cartesian coordinate system: the squared distance between two points equals the sum of squares of the difference in each coordinate between the places. The Indian Baudhayana Sulba-sutra, who was authored between 800 and 400 BCE, makes reference to this theorem. Nevertheless, Pythagoras eventually received credit for the theorem. Additionally, it is Euclid’s Elements statement number 47 from Book I. There have been a plethora of original Pythagorean Theorem Formula proofs and extensions developed. Euclid himself demonstrated, using extensions as a starting point, that any symmetrical regular figures drawn on the sides of a right triangle meet the Pythagorean relationship: the figure placed on the hypotenuse has an area equal to the sum of the figures drawn on the legs.

The theorem can be generalised in several ways, including higher-dimensional spaces, non-Euclidean spaces, objects that are not right triangles, and objects that are not triangles at all but n-dimensional solids. Outside of mathematics, the Pythagorean Theorem Formula has sparked attention as a symbol of mathematical obscurity, mystique, or intellectual strength; popular references can be found in literature, plays, musicals, songs, stamps, and cartoons.

Some mathematical ideas from trigonometry, the study of triangles, are required to better understand certain problems involving aircraft and propulsion. It is important to understand the definition and proof of the Pythagoras Theorem. Taking an example of a right triangle is necessary. A right triangle is a three-sided figure with one 90-degree angle. A right angle is a 90-degree angle, and it is from this angle that the right triangle gets its name. The hypotenuse, h, is the side of the triangle opposite the right angle. It is the longest of the right triangle’s three sides. Because this is the longest side, the term “hypotenuse” is derived from two Greek words that mean “to stretch”.

### Pythagoras Theorem Formula

The Pythagorean Theorem Formula states that the squares on the hypotenuse (the side across from the right angle) of a right triangle, or, in standard algebraic notation, a2 + b2, are equal to the squares on the legs. Although Pythagoras (c. 570–500/490 BCE), a Greek mathematician and philosopher, has traditionally been linked to the theorem, it actually dates back far more. Four Babylonian tablets from between 1900 and 1600 BCE show some understanding of the theorem and provide an extremely precise square root of 2 calculations.

### Derivation of Pythagoras Theorem Formula

It is important to go through the derivation of Pythagorean Theorem Formula. To solve questions, understanding the derivation of the Pythagorean Theorem Formula. Students who are facing challenges in understanding the derivation of the Pythagorean Theorem Formula can take guidance from the teachers at Extramarks. The study resources available on Extramarks are regularly updated and very credible. Using the NCERT solutions, students can easily get exact solutions to questions.

### Pythagoras Theorem Proof

Pythagoras extended the conclusion to any right triangle. The theorem can be proved in a variety of algebraic and geometric ways. The majority of these start with a square construction on a sketch of a basic right triangle. The squares drawn on the three sides of the triangle are depicted in the figure at the top of this page. A square is a special case of a rectangle with all of its sides being the same length. There are numerous proofs for this most fundamental of all geometric theorems. The theorem can also be generalised from a plane triangle to a trirectangular tetrahedron, and is known as de Gua’s theorem in that case. The various Pythagorean theorem proofs all appear to require the application of some version or consequence of the parallel postulate: proofs by dissection rely on the complementarity of the acute angles of the right triangle, proofs by shearing rely on explicit parallelogram constructions, proofs by similarity require the existence of non-congruent similar triangles, and so on. Based on this observation, Brodie demonstrated that the parallel postulate corresponds to the Pythagorean theorem.

### Converse of Pythagoras Theorem

The Pythagoras theorem’s opposite, known as the converse, can be used to determine whether a triangle is acute, right, or obtuse by comparing the sum of its two sides to the square of its third side. The most common trigonometric theorem is the Pythagorean theorem.

According to the converse of Pythagoras’ theorem, a triangle is a right triangle if the square of its longest side equals the sum of its squares on the other two sides. The Pythagoras theorem is completely reversed in the opposite. The main use of the Pythagorean theorem’s converse is that measurements can be used to determine the kind of triangle right, acute or obtuse. Constructing the triangle is straightforward once it has been found. Three situations can happen:

1. A triangle is a right triangle if its two sides’ square sums are regarded as equal to the square of the hypotenuse.
2. A triangle is an obtuse triangle if the sum of the squares of its two sides is less than the square of its hypotenuse.
3. A triangle is an acute triangle if the sum of the squares of its two sides is greater than the square of its hypotenuse.

### Formula and Proof of Converse Pythagoras Theorem

Students are advised to learn the converse Pythagorean Theorem Formula along with its proof.

The Pythagorean Theorem Formula is an important topic in Mathematics. Students will be able to prepare effectively for their examination by practising questions with the Pythagorean Theorem Formula. Solving questions consistently by using the Pythagorean Theorem Formula helps students in gaining a thorough understanding of Pythagoras theorem. It is necessary for students to get a deeper understanding of the Pythagorean Theorem Formula.

### Applications of Pythagoras Theorem

Several examples of how the Pythagoras theorem is applied include

1. The Pythagorean theorem is frequently used to determine the lengths of a right-angled triangle’s sides.
1. To determine the diagonal length of a rectangle, square, etc., use the theorem.
1. In trigonometry, Pythagoras’ theorem is used to determine trigonometric ratios such as sin, cos, tan, cosec, sec, and cot.
1. For face identification in security cameras, Pythagoras’ theorem is applied.
1. In the disciplines of engineering and building, architects apply the Pythagoras theorem approach.
1. Applying the Pythagoras theorem to mountain surveying
2. Finding the shortest path is another application for it in navigation.
3. We can get the formula for base, perpendicular, and hypotenuse using the Pythagoras theorem.
4. With the use of the Pythagoras theorem, painters utilise ladders to paint on tall structures.
5. The theorem is used to determine how steep a mountain’s or a hill’s slope is.
6. A triangle’s rightness is determined using the converse of Pythagoras theorem.

### Solved Examples on Pythagoras Theorem

Students need to keep practising questions that are specific to the Pythagorean Theorem Formula. All the questions based on the Pythagorean Theorem Formula should be practised from time to time. To get well-structured solutions related to the Pythagorean Theorem Formula take help from the NCERT solutions. All the important questions regarding the Pythagorean Theorem Formula can be easily solved using the NCERT solutions.