Pythagoras Theorem | Definition & History
is an important topic in Maths, which explains the relation between the sides of a right-angled triangle. It is also sometimes called the Pythagorean Theorem. The formula and proof of this theorem are explained here with examples about Pythagoras Theorem | Definition & History.
Pythagoras theorem is basically used to find the length of an unknown side and angle of a triangle. By this theorem, we can derive base, perpendicular and hypotenuse formula. Let us learn mathematics of Pythagorean theorem in detail here.
Table of Contents:
- Statement
- Formula
- Proof
- Applications
- Problems
Pythagoras Theorem Statement
Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“. The sides of this triangle have been named as Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°. The sides of a right triangle (say a, b and c) which have positive integer values, when squared, are put into an equation, also called a Pythagorean triple.
History
The theorem is named after a greek Mathematician called Pythagoras.
Pythagoras Theorem Formula
Consider the triangle given above:
Where “a” is the perpendicular side,
“b” is the base,
“c” is the hypotenuse side.
According to the definition, the Pythagoras Theorem formula is given as:
| Hypotenuse2 = Perpendicular2 + Base2 c2 = a2 + b2 |
The side opposite to the right angle (90°) is the longest side (known as Hypotenuse) because the side opposite to the greatest angle is the longest.
Consider three squares of sides a, b, c mounted on the three sides of a triangle having the same sides as shown.
By Pythagoras Theorem –
Area of square A + Area of square B = Area of square C
Example
The examples of theorem based on the statement given for right triangles is given below:
Consider a right triangle, given below:
Find the value of x.
X is the side opposite to right angle, hence it is a hypotenuse.
Now, by the theorem we know;
Hypotenuse2 = Base2 + Perpendicular2
x2 = 82 + 62
x2 = 64+36 = 100
x = √100 = 10
Therefore, we found the value of hypotenuse here.
Pythagoras Theorem Proof
Given: A right-angled triangle ABC, right-angled at B.
To Prove- AC2 = AB2 + BC2
Construction: Draw a perpendicular BD meeting AC at D.
Proof:
We know, △ADB ~ △ABC
Therefore, ADAB=ABAC (corresponding sides of similar triangles)
Or, AB2 = AD × AC ……………………………..……..(1)
Also, △BDC ~△ABC
Therefore, CDBC=BCAC (corresponding sides of similar triangles)
Or, BC2= CD × AC ……………………………………..(2)
Adding the equations (1) and (2) we get,
AB2 + BC2 = AD × AC + CD × AC
AB2 + BC2 = AC (AD + CD)
Since, AD + CD = AC
Therefore, AC2 = AB2 + BC2
Hence, the Pythagorean theorem is proved.
Note:Pythagorean theorem is only applicable to Right-Angled triangle.
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