Math Help: Introduction to the PythagoreanTheorem
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.
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
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.
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=?
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
c2 = b X DC …………….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
a2 = b X BD …………….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)
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.
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 a=3, c=4 then from the theorem 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.