Mr. James Abram Garfield proposed this proof in 1876. The surprising part is that Mr. Garfield was a politician! And not an ordinary one, he became the 20th president of the US. Wow!
He proved the Pythagoras theorem with the help of geometrical construction and formulas for the area of triangle and trapezium. So you should already know the following formulas to understand the proof:
- Area of a Triangle =
- Area of a Trapezium =
Any grade 8 student should be able to understand this proof. Let us see how he did it!
Statement of Pythagoras Theorem
Pythagoras theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.
AB2 + AC2 = BC2
As AB = a, AC = b and BC = c, we have to prove:
a2 + b2 = c2
Step 1 – A Simple Construction
We create a copy of the right triangle and rotate it by 90 degrees in the clockwise direction.
Next, we move up the triangle to align line segments AB and BD. So we form a straight line segment AD.
Step 2 – Find angle ∠5
In the triangle ABC:
∠1 + ∠2 + ∠3 = 180
∠1 + 90 + ∠3 = 180
∠1 + ∠3 = 180 – 90 = 90
As AD is straight line segment, we can say:
∠1 + ∠5 + ∠3 = 180
∠5 + ∠1 + ∠3 = 180
We can replace ∠1 + ∠3 by 90
∠5 + 90 = 180
∠5 = 90
We will use this later in our proof. Next we join E and C to form trapezium ADEC ( We know that DE is parallel to AC. How? )
Step 3 – Area of Trapezium ADEC
The formula for the area of trapezium is:
Here parallel sides are DE and AC. So sum of parallel sides = a + b.
Distance between the parallel sides is AD = AB + BD = a + b. Using these values in the formula we get:
Step 4 – Area of Trapezium ADEC from Triangles
Three right triangles – ABC, BDE, and BCE form the trapezium. So its area is the sum of areas of these triangles.
Area of Trapezium = area of triangle ABC + area of triangle BCE + area of triangle BDE
Notice that we used ∠5 = 90 degrees in calculating the area of the triangle BCE.
Step 5 – Equating the Areas
In the end, we equate the areas obtained in the previous two steps.
Multiplying both sides by 2:
a2 + 2ab + b2 = c2 + 2ab
Subtracting 2ab from both sides, we get:
a2 + b2 = c2
We have proved the theorem!
There are many ways to prove the Pythagoras Theorem, however we chose one of the simplest methods.