As you know, a right-angled triangle (a special classification in the triangle family) has one angle equal to 90 degrees and the sum of the other two angles is 90 degrees. The old is gold concept that has accompanied the topic of the right-angled triangle is the Pythagorean Theorem. This serves as an important part of Euclidian geometry that efficiently describes the relationship among the sides of a right triangle by showing the interrelationship between the base, perpendicular, and hypotenuse of a right triangle.
The concept is mainly and primarily used to find one unknown side of the right triangle when the other two sides are known.
The concept also has a converse relation to check out whether the triangle is a right-angled triangle or not.
Pythagorean Theorem
Basically, the Pythagoras theorem has its origin actually 2500 years ago and is named after a Greek mathematician Pythagoras. The theorem in simple words states that in a right-angled triangle the sum of the square of base and the square of height (altitude) is equal to the square of the hypotenuse.
Formula: Hypotenuse ^2 = Base^2+ height^2
The basic terms used in Pythagoras theorem are:
- Base
- Perpendicular / height/ altitude
- Hypotenuse (the side opposite the 90-degree angle and also the largest side of the triangle.
Pythagorean triplets: Any three numbers that can satisfy the relation of the Pythagoras theorem (hypotenuse^2= base^2+ height^2) are called Pythagorean triplets. Some examples of Pythagorean triplets are (3, 4, 5);(5, 12, 13);(8, 15, 17) etc.
The converse of the Pythagoras theorem states that “if the square of one side is equal to the sum of squares of other two sides then the triangle is a right-angled triangle and the angle opposite to the first (longest) side is a right angle, i.e. 90 degrees.
Proof of Pythagoras theorem
The proof of the Pythagoras theorem also is based on the concept of similarity of triangles.
Given: A triangle PQR that is right-angled at Q. PR is the hypotenuse, QR is the base and PQ is the height of the triangle.
To prove that: hypotenuse^2 = base^2 * height^2
Construction: Draw a perpendicular from Q to meet hypotenuse PR at D.
Proof: Taking triangle RDQ and triangle RQP similar, we get RD/QR=QR/PR
Cross-multiplying denominators we get, PR^2=PQ*QR_______ (1)
Also, triangles QDP and RQP are similar, so we get PQ^2= PD*RP____ (2)
Adding equations 1 and 2 we get PR^2=PQ^2+QR^2
Hence Pythagoras theorem is well proved.
*In Cuemath this is well explained with even the minor details.
Applications of Pythagorean Theorem
- The technique is widely applied at construction sites by engineers for architectural purposes.
- This is used in finding the shortest route during navigation.
- The theorem is used to calculate the steepness of slopes of hills and mountains.
- Pythagorean Theorem has wider applicability in trigonometry, which purely has the concept of triangles (right-angled) as its base. The trigonometric ratios absolutely depend on the theorem.
- Nowadays this is used in security cameras for face recognition.
- Though the theorem is used to find the sides of right-angled triangles it also has a wider application in finding the length of diagonals of squares, etc.
- While surveying mountains or very tall buildings the theorem finds perfect applicability.
With so many above-mentioned applications the Pythagoras theorem has some more applications in day-to-day life too. This has actually proved to be a blessing of mathematics to mathematicians and others too. In real life this can be used to find out the distance between planes, calculating surface areas, and volumes, etc.
In the article, we learned about the Pythagoras theorem, its converse, derivation of Pythagoras theorem, important and usual applications of Pythagoras theorem.
The Pythagoras theorems find their applications only in the case of right-angled triangles. Trigonometry and Pythagoras theorem take only right-angled triangles as their source.