The Pythagoras theorem which is also sometimes referred the Pythagorean theorem is the most important formula of a geometry branch. According to Pythagoras, the square of the hypotenuse is equal to the sum of the squares of the other two sides of a triangle. In this lesson, you will learn about the Pythagoras theorem, its derivations, and equations followed by solved real-world problems on the Pythagoras theorem triangle and squares.
What Is Pythagoras Theorem?
The Pythagoras theorem states that if a triangle is right-angled (90 degrees), then the square of the hypotenuse is equal to the sum of the squares of the other two sides. In the given triangle ABC, we have BC2 = AB2 + AC2. Here, AB is the base, AC is the altitude or the height, and BC is the hypotenuse.
Pythagoras Theorem Equation
Pythagoras theorem equation helps you to solve right-angled triangle problems, using the Pythagoras equation: c2 = a2 + b2 ('c' = hypotenuse of the right triangle whereas 'a' and 'b' are the other two legs.). Hence, any triangle with one angle equal to 90 degrees will be able to produce a Pythagoras triangle. We can use this Pythagoras equation: c2 = a2 + b2 there.
History of Pythagoras Theorem
Pythagoras theorem was introduced by the Greek Mathematician Pythagoras of Samos. He was an ancient Ionian Greek philosopher. He started a group of mathematicians who works religiously on numbers and lived like monks. Finally, the Greek Mathematician stated the theorem hence it is called by his name as the "Pythagoras theorem." Though it was introduced many centuries ago its application in the current era is obligatory to deal with pragmatic situations.
Although Pythagoras introduced and popularised the theorem, there is sufficient evidence proving its existence in other civilizations, 1000 years before Pythagoras was born. The oldest known evidence dates back to between 20th to 16th Century B.C in the Old Babylonian Period.
Pythagoras Theorem Formula
The Pythagoras theorem formula states that in a right triangle ABC, the square of the hypotenuse is equal to the sum of the square of the other two legs. If AB, BC, and AC are the sides of the triangle, then: BC2 = AB2 + AC2. While if a, b, and c are the sides of the triangle, then c2 = a2 + b2. In this case, we can say that AB is the base, AC is the altitude or the height, and BC is the hypotenuse.
Pythagoras Theorem Proof
Pythagoras theorem can be proved in many ways. Some of the most common and most widely used methods are by using the algebraic method proof and using the similar triangles method to solve them. Let us have a look at both of these methods individually in order to understand the proof of this theorem.
- Algebraic Method Proof of Pythagoras Theorem
- Pythagoras Theorem Proof using Similar Triangles
Algebraic Method Proof of Pythagoras Theorem
Algebraic method proof of Pythagoras theorem will help us in deriving the proof of the Pythagoras Theorem by using the values of a, b, and c (values of the measures of the side lengths corresponding to sides BC, AC, and AB respectively). Consider four right triangles ABC where b is the base, a is the height and c is the hypotenuse. Arrange these four congruent right triangles in the given square, whose side is a + b.
The area of the square so formed by arranging the four triangles is c2. The area of a square with side (a + b) = Area of 4 triangles + Area of square with side c. This implies (a + b)2 = 4×1/2×(a×b)+ c2, a2 + b2 + 2ab = 2ab + c2. Therefore, a2 + b2 = c2. Hence Proved.
Pythagoras Theorem Proof using Similar Triangles
Two triangles are said to be similar if their corresponding angles are of equal measures and their corresponding sides are in the same ratio. Also, if the angles are of the same measure, then we can say by using the sine law, that the corresponding sides will also be in the same ratio. Hence, corresponding angles in similar triangles will lead us to equal ratios of side lengths.
In triangle ABD and triangle ACB:
- ∠A = ∠A (common)
- ∠ADB = ∠ABC (both are right angles)
Thus, triangle ABD and triangle ACB are equiangular, which means that they are similar by AA similarity criterion. Similarly, we can prove triangle BCD similar to triangle ACB. Since triangles ABD and ACB are similar, we have AD/AB = AB/AC. Thus we can say that AD × AC = AB2. Similarly, triangles BCD and ACB are similar. That gives us CD/BC = BC/AC. Thus, we can also say that CD × AC = BC2. Now, using both of these similarity equations, we can say that AC2 = AB2 + BC2. Hence Proved.
Pythagoras Theorem Triangles
Right triangles follow the rule of the Pythagoras theorem and they are called Pythagoras theorem triangles. The length of all the three sides are being collectively called Pythagoras triples. For example, 3, 4, and 5 can be called as one of the sets of such triangles. There are a lot more right-angled triangles which are called Pythagoras triangles. All such triangles follow one common rule: c2 = a2 + b2.
Pythagoras Theorem Squares
As per the Pythagorean theorem Hypotenuse2 = Perpendicular2 + Base2 or c2 = a2 + b2, which further justifies that the area of the square built upon the hypotenuse of a right triangle will be equal to the sum of the area of the squares built upon the other two sides. And these squares are known as Pythagoras squares.
Applications of Pythagoras Theorem
Though it is necessary to learn the basic concepts such as theorem statements and their mathematical representation, we would be more curious in understanding the applications of the Pythagoras theorem which we face in day-to-day life situations.
Here are some of the applications of the Pythagoras Theorem
- Engineering and Construction fields
Most architects use the technique of the Pythagorean theorem to find the value as well as when length or breadth are known it is very easy to calculate the diameter of a particular sector. It is mainly used in two dimensions in engineering fields.
- Face recognition in security cameras
We are more familiar with face recognition nowadays it reduces the turmoil in investigating the crimes in the security areas. It undergoes the concept of the Pythagorean theorem that is, the distance between the security camera and the place where the person is noted is well projected through the lens using the concept.
- Woodworking and interior designing
As the main concept indicates if the cardboards being square can be made into a triangle easily by cutting diagonally then very easily the Pythagoras concept can be applied. Most woodworks are made on the strategy which makes it easier for designers to proceed.
It's a very amazing fact but people traveling in the sea use this technique to find the shortest distance and route to proceed to their concerned places.
Usually, surveyors use this technique to find the steep mountainous region, knowing the horizontal region it would be easier for them to calculate the rest using the Pythagoras concept. The fixed distance and the varying one can be looked through a telescope by the surveyor which makes the path easier.
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Examples on Pythagoras Theorem
Example 1: Consider a right-angled triangle. The measure of its hypotenuse is 16 units. One of the sides of the triangle is 8 units. Find the measure of the third side using the Pythagoras theorem formula?
Given : H = 16 units
Let us consider the other given side of a triangle as perpendicular height = 8 units
Let us apply the given dimensions to the Pythagoras theorem formula
Hypotenuse2 = Base2 + Height2
162 = B2 + 82
B2 = 256 - 64
B = √192 = 13.856 units
Therefore, the measure of the third side of a triangle is 13.856 units.
Example 2: Julie wanted to wash her building window which is 12 feet off the ground. She has a ladder that is 13 feet long. How far should she place the base of the ladder away from the building?
We can visualize this scenario as a right triangle. We need to find the base of the right triangle formed. We know that, Hypotenuse2 = Base2 + Height2. Thus we can say that b2 = 132 - 122 where b is the distance of the base of the ladder from the feet of the wall of the building. Hence, we get b = 5.
Therefore, the base of the ladder is 5 feet away from the building.
Example 3: Kate, Jack, and Noah were having a party at Kate's house. After the party gets over, both went back to their respective houses. Jack's house was 8 miles straight towards the east, from Kate's house. Noah's house was 6 miles straight south from Kate's house. How far away were their houses (Jack's and Noah's)?
We can visualize this scenario as a right-angled triangle. That means Jack and Noah are hypotenuses apart from each other. So the distance between Jack's house and Noah's house can be found by using the Pythagoras theorem: d2 = 82 + 62 = 100, where d is the distance of the house. Upon solving, we get d = 10.
Therefore, the houses are 10 miles away from each other.
FAQs on Pythagoras Theorem
What Is the Pythagoras Theorem in Math?
Right triangles follow the rule of the Pythagoras theorem and they are called Pythagoras theorem triangles. All such triangles will follow just one common rule: c2 = a2 + b2.
What is the Converse of Pythagoras theorem?
The converse of Pythagoras theorem is: If the sum of the squares of any two sides of a triangle is equal to the square to the third (largest) side, then it is said to be a right-angled triangle.
What Is the Use of the Pythagoras Theorem?
The Pythagoras theorem works only for right-angled triangles. When any two values are known, we can apply the Pythagoras theorem and calculate the other.
How Do You Find the Pythagoras Theorem for a Right Triangle?
The square of the hypotenuse of a right triangle is equal to the sum of the square of the other two sides. When any two values are known, we can apply the theorem and calculate the other.
What Are the Applications of the Pythagorean Theorem in Real Life?
The Pythagorean theorem helps in
- Computing the distance between points on the plane.
- Calculating the perimeter, the surface area, the volume of geometrical shapes, and so on.
- In architecture and construction industries.
- In surveys
How Do You Solve A Squared Plus B Squared Equals to C Squared Using Pythagorean Theorem?
A squared plus B squared equals to C squared implies c2 = a2 + b2, which is the formula of the Pythagoras theorem, for figuring out the hypotenuse in a right triangle when the other two sides are given.
Can We Apply the Pythagorean Theorem to Any Triangle?
No, you can't apply the Pythagoras or the Pythagorean theorem to any triangle. It needs to be a right-angled triangle only then one can use the Pythagoras theorem and obtain the relation where the sum of two squared sides is equal to the square of the third side.