If you've taken any geometry or trigonometry you've probably become familiar with the Pythagorean Theorem. Get some help understanding the concept behind the theorem with these examples.

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Around 2530 years ago, Pythagoras first created the Pythagorean Theorem. A simple Pythagorean Theorem proof is making a pyramid with a perfect square or rectangular base.

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### The Theorem

*The square of the hypotenuse of a right angled triangle is equal to the sum of squares of other two sides.*See the attached snap of the triangle below (Click on it to enlarge):

So, as per Pythagoras theorem we can write that:

**(BC)**^{2 }= (AB)^{2 }+ (AC)^{2}If we consider the side

**AB**as**a, AC**as**c**and**BC**as**b**then we can write the Pythagorean Theorem as*b2=a2+ c2*Now you are in a position to calculate the length of any one side of a right angled triangle, if the lengths of the other two sides are given.

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### Practice Problems

Solve the following examples. Assume

**c**is the hypotenuse of a right angled triangle and**a, b**are the other two sides of the same triangle:a = 3; b = 4; c=?

a=5; c= 13; b=?

a=8; b=15; c=?

a= 9; c=41; b=?

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### Pythagorean Theorem Proof

Pythagoras' Theorem has more than 300 proofs. The simplest proof of the theorem is based on

**the similar triangles concept**:- Take the triangle ABC with AB=a, AC=c and BC=b.
- Drop a line from A to D which is perpendicular to BC.
- Consider AD=d.
- Triangle ACD is similar to the triangle ABC as:

Angle ADC = Angle CAB = 90 degree

Angle ACD = Angle ACB

Side AC is common for both the triangle

So, we can write from similar triangle principle:

**c / b = DC / c****c**^{2 }= b X DC โฆโฆโฆโฆโฆ.eqn.1- Again, triangle ABD and triangle ABC are similar because:

Angle ADB = Angle CAB = 90 degree

Angle ABD = Angle ABC

Side AB is common for both triangles.

So, we can write:

**a / b = BD / a****a**^{2 }= b X BD โฆโฆโฆโฆโฆ.eqn.2-
From
**eqn.1**and**eqn.2**we can write:

**c2 +a2 = (b X DC) + (b X BD)****= b X (DC + BD)****= b X BC(***as, BC consists of*DC and BD)**= b X b(as, we already assume BC=b)****= b2** - slide 5 of 6
### Real Life Examples

- Making a perfect rectangular basketball and volleyball court.
- Measuring the height of ramp.
- Calculating distance between two points if co-ordinates of the points are given.

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### Pythagoras Triples

Pythagoras triples are sets of three integer numbers which follows Pythagoras' Theorem. For example, take 3, 4, 5.

Remember the Pythagoras theorem (

*b2=a2+ c2).*Now, if you take

then from the theorem*a=3, c=4***b**will be equal to 5. There are many such sets of integers like this: 5, 12,13; 17, 24, 25; 9, 40, 41 etc.People have been using these triples even before human beings learned to write.