## Pythagoras
He was credited with many discoveries in the field of science, mathematics, music, astronomy, and medicine. The discoveries done by him are, ## Pythagoras or Pythagorean TheoremPythagoras theorem is based on the right-angled triangle or right triangle only. The theorem states that In other words ## Components of Right TriangleThe following figure represents a right triangle ∆ABC. **Base:**It is a side of the right triangle that is adjacent to the perpendicular and hypotenuse. In ∆ABC,**AB**is base.**Perpendicular:**It is a side of the right triangle that is adjacent to the base and hypotenuse. It is called the height of the triangle. In ∆ABC,**AC**is perpendicular.**Hypotenuse:**The side opposite to the right angle is called the hypotenuse. In other words, the longest side of the right triangle is called the hypotenuse. In ∆ABC,**BC**is the hypotenuse.**Right-angle:**In geometry, the right-angle is an angle that makes an angle of 90°. In ∆ABC,**∠A**is a right-angle .
## Pythagoras TriplesPythagoras or Pythagorean triples is a set of three positive integers that satisfies the Pythagoras theorem. The least Pythagorean triple is In ∆ABC, a ^{2}+b^{2}=c^{2}The following table enlists some Pythagorean triples.
- It always has one even number in a triple.
- The value of c will always be odd.
- It may have two prime numbers.
## Pythagoras Theorem FormulaConsider the following figure. In ∆ABC, AC is perpendicular or height, AB is base, and BC is the hypotenuse. The length of perpendicular, base, and hypotenuse is a, b, and c, respectively. According to the Pythagoras theorem, the Pythagoras theorem formula can be written as: Perpendicular ^{2} + Base^{2} = Hypotenuse^{2}Or AC ^{2}+ AB^{2} = BC^{2}Or a ^{2}+ b^{2} = c^{2}## Pythagoras Theorem Proof
## Proof 1:In the following figure, we have drawn a perpendicular (BD) from point B that meets at point D on the hypotenuse. The perpendicular divides the triangle into two triangles, i.e., ∆ADB and ∆BDC.
According to the above statement, ∆ABC=∆ADB Adding the equations (i) and (ii), we get: AB AB ^{2}+BC^{2}=AC^{2}Hence, the Pythagoras theorem is proved. Let's see the second way to prove the theorem. ## Proof 2:In the following figure, we have drawn a square Now, we will find the area of both squares and triangles, separately. We know that, the Area of square ABCD=(a+b) We know that, Area of a triangle=ab There are a total of four triangles, so the area of four triangles will be: Area of four triangles= 4×ab= Area of square EFGH=c The total area of the square ABCD will be:
Putting the values, we get: (a+b) Cancel out the 2ab on both sides, we get: a ^{2}+b^{2}=c^{2}
## Pythagoras Theorem Problems
Given, AB = 12 cm, BC = 5 cm, AC = 13 cm According to the Pythagoras theorem, 13
In the ∆ABC, given that BC = 3 cm and AB = 4 cm. According to the Pythagoras theorem, Putting the values of AB and BC in the above formula, we get: 3
In the ∆ABC, given that AC = 10 cm and BC = 8 cm. According to the Pythagoras theorem, Putting the values of AC and BC in the above formula, we get: 10
In the ∆ABC, given that AC = 50 m and AB = 30 m. According to the Pythagoras theorem, Putting the values of AC and AB in the above formula, we get: 50
In the above figure, we see that there are two triangles ∆ABC and ∆ADC. Let's take the triangle ∆ABC and find the diagonal. According to the Pythagoras theorem, AC
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