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Pythagorean theorem: a2 + b2 = c2
Let's look at an easy, visual proof of the Pythagorean theorem.
(If the drawing canvas does not appear, is really short, or is blank, reload the page.
The drawing canvas can be resized as you please (except on multitouch devices)
by dragging and releasing the magenta drag-handle at the lower right of the canvas; the demo will restart.)
Consider a right triangle with side lengths a<b<c (hypoteneuse) and
angles A<B <C=90 degrees.
Angle A is opposite side a. Angle B is opposite side b.
Let's show angle A with the color amber and angle B with blue
(amber = A; blue = B). The amber angle A plus the blue angle B equals
90 degrees because the angles of a triangle sum to 180 degrees and
C is 90 degrees.
The Pythagorean theorem states that a2 + b2 = c2.
For each side of the triangle, the drawing to the left also shows a square.
The area of the red square is a2.
The area of the green square is b2.
The area of the gray square is c2.
The Pythagorean theorem says that the gray square area is equal to
the sum of the red square area and the green square area.
Let's do something with the gray square to see if we can prove this.
Let's clone a copy of the triangle and slide it to the left and upwards
until it touches the side of the square.
The side of the square is the same length (c)
as the hypoteneuse of the triangle because that's the size we made the gray square.
What next? Can we tell anything about the gray angle between the
triangle's side a and the side of the square: the angle adjacent to
the triangle's blue angle B?
All corners of any square are 90 degrees so that gray angle must be
90 - B. So it must equal the amber angle A since A + B = 90 degrees!
Let's clone another copy of the triangle, rotate it 90 degrees
counterclockwise, and slide it adjacent to the first cloned triangle and
the side of the square. It fits perfectly: no gap!
Similar to the previous step,
let's clone a third copy of the triangle, rotate it 180 degrees
counterclockwise, and slide it adjacent to the second triangle and
the side of the square. It fits fine!
Would a fourth clone of the triangle fit adjacent to the third cloned triangle,
the side of the square, and the first cloned triangle?.
Yes, each corner of the square is exactly covered by one amber angle A
from one triangle and one blue angle B from another triangle.
Everything fits perfectly!
The four triangles don't completely cover the gray square.
A small square at the center is left over; what size is it?
Looking at the triangles, we see that the sides of the small square have length b-a.
Now, the area of the large square (c2)
equals the area of four triangles (4*ab/2 = 2ab) plus the area of the small square (b-a)2.
So c2 = 2ab + (b2 -2ab + a2).
2ab and -2ab cancel out. c2 = b2 + a2,
or a2 + b2 = c2.
This is the Pythagorean theorem!
We chose a<b and the four triangles didn't completely cover the gray square.
What about the (isosceles) right triangle with a = b?
We would find that the four triangles completely cover the large gray square
and there is no small square left over in the middle.
Each triangle's area would be one half a2.
Four triangles' area would equal the large gray square:
2*a2 = a2 + b2 = c2 (for a=b).
Instead of placing the triangles inside the large gray square,
we could have placed them outside it such that the large gray square
plus four triangles exactly forms a new larger square with sides
of length a+b. Can you do this?
Can you use a little algebra to prove the Pythagorean theorem?
Changing direction, let's look at a special triangle.
Suppose we draw a triangle with side lengths 3, 4, and 5.
Can we tell whether this is a right triangle or not?
32 + 42 = 9 + 16 = 25 = 52.
This is a right triangle! Take a sheet of paper with a square corner.
Measure off three units (say three inches or centimeters) horizontally
from the corner and mark that point.
Measure off four units vertically
from the corner and mark that point.
Measure the distance between the two points: is it five units?
The 3-4-5 right triangle is easy to remember
and can be useful for measuring whether a corner of a room is square (90 degrees).
Do the sides 5-12-13 form a right triangle?
Integers that form a right triangle are called
The Pythagorean theorem has been known for at least 2300 years.
Pythagorean triples may have been known for 4000 years.
Our proof used algebra but there are many different proofs of the Pythagorean theorem using only geometry.
I hope that you found this outline of a proof to be simple and memorable.
Further info: Wikipedia article: Pythagorean Theorem.
Note: the Pythagorean theorem applies only for right triangles in a plane.
It does not apply for right triangles on a non-Euclidean surface such as
on a sphere. Take a ball and draw a large right "triangle" on it.
Measure that it is different than in a plane.
For example, legs of 3 and 4 do not give 5 for the "hypoteneuse" on the ball!
I hope you found this interesting, useful, and/or fun.
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