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Pythagoras Theorem
Pythagoras Theorem 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. This theorem is basically used for the right-angled triangle and by which we can derive base, perpendicular and hypotenuse formula. Let us learn this 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.
Pythagoras Theorem-Right Angle Triangle
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.
Pythagoras Theorem
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:
Pythagoras theorem Example
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.
Pythagoras Theorem Proof
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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Applications of Pythagoras Theorem
To know if the triangle is a right-angled triangle or not.
In a right-angled triangle, we can calculate the length of any side if the other two sides are given.
To find the diagonal of a square.
Useful For
Pythagoras theorem is useful to find the sides of a right-angled triangle. If we know the two sides of a right triangle, then we can find the third side.
How to use?
To use this theorem, remember the formula given below:
c2 = a2 + b2
Where a, b and c are the sides of the right triangle.
For example, if the value of a = 3 cm, b = 4 cm, then find the value of c.
We know,
c2 = a2 + b2
c2 = 32+42
c2 = 9+16
c2 = 25
c = √25
c = 5
Hence, the third side is 5 cm.
As we can see, a + b > c
3 + 4 > 5
7 > 5
Hence, c = 5 cm is the hypotenuse of the given triangle.
How to find whether a triangle is a right-angled triangle?
If we are provided with the length of three sides of a triangle, then to find whether the triangle is a right-angled triangle or not, we need to use the Pythagorean theorem.
Formula of mode , median, and mean.
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