Pythagoras Theorem
Contents
What is the Pythagorean Theorem?
What is the Pythagorean Theorem?
The Pythagoras Theorem is also referred to as the Pythagorean Theorem
Pythagorean Theorem is used to find a side of any right triangle. It is 
, where 
and 
are the legs of the triangle, and 
is the hypotenuse.
Why is it useful?
To find sides and angles of right triangles. Also, Trigonometry is pointless without it. If you know three angles of a triangle you can use the Pythagorean Theorem to find the sides or the area even if the angles are not right. It is probably the most famous Theorem in all of math!
Can we prove it?
Yes! The are hundreds of proves. I will just show you a few of them. Mathematicians even make a hobby of finding these proves. Even a US president made a published proof! Of, course this was a president in the 1800s because, well, presidents now are not really up to proving something like that. (You know, Trump and the others).
Proofs
Proof 1
We use ![$[ABC]$](http://latex.artofproblemsolving.com/d/3/3/d33cc80fa8f093e155c5be46d2e5d9da3d7e1ef5.png)
to denote the area of triangle 
.
Let 
be the perpendicular to side 
from 
.
![[asy] pair A, B, C, H; A = (0, 0); B = (4, 3); C = (4, 0); H = foot(C, A, B); draw(A--B--C--cycle); draw(C--H); draw(rightanglemark(A, C, B)); draw(rightanglemark(C, H, B)); label("$A$", A, SSW); label("$B$", B, ENE); label("$C$", C, SE); label("$H$", H, NNW); [/asy]](http://latex.artofproblemsolving.com/a/1/9/a19bed9f5ac971139756a395da4e29366d45fc52.png)
Since 
are similar right triangles, and the areas of similar triangles are proportional to the squares of corresponding side lengths,
![$\frac{[ABC]}{AB^2} = \frac{[CBH]}{CB^2} = \frac{[ACH]}{AC^2}$](http://latex.artofproblemsolving.com/a/9/b/a9b69ba61346d3e73a308b035c9372840821fd6b.png)
.
But since triangle is composed of triangles 
and 
, ![$[ABC] = [CBH] + [ACH]$](http://latex.artofproblemsolving.com/8/f/f/8ff368e96540b4d60150bc7163f9cb6446017a54.png)
, so 
. ∎
Proof 2
Consider a circle 
with center 
and radius 
. Since and 
are perpendicular, is tangent to . Let the line meet at 
and 
, as shown in the diagram:
Evidently, 
and 
. By considering the power of point 
with respect to , we see
. ∎
Proof 3
and 
are squares.
![[asy] pair A, B,C,D; A = (-10,10); B = (10,10); C = (10,-10); D = (-10,-10); pair E,F,G,H; E = (7,10); F = (10, -7); G = (-7, -10); H = (-10, 7); draw(A--B--C--D--cycle); label("$A$", A, NNW); label("$B$", B, ENE); label("$C$", C, ESE); label("$D$", D, SSW); draw(E--F--G--H--cycle); label("$E$", E, N); label("$F$", F,SE); label("$G$", G, S); label("$H$", H, W); label("a", A--B,N); label("a", B--F,SE); label("a", C--G,S); label("a", H--D,W); label("b", E--B,N); label("b", F--C,SE); label("b", G--D,S); label("b", A--H,W); label("c", E--H,NW); label("c", E--F); label("c", F--G,SE); label("c", G--H,SW); [/asy]](http://latex.artofproblemsolving.com/2/e/7/2e79de6673cd591bf38bff3efa048140bda61e9f.png)
. ∎
Pythagorean Triples
Pythagorean Triples are a group of integers a,b and c in which . These are the first few:
(3,4,5) (5,12,13) (7,24,25) (8,15,17)
(9,40,41) (11,60,61) (12,35,37) (13,84,85)
(15,112,113) (16,63,65) (17,144,145) (19,180,181)
(20,21,29) (20,99,101) (21,220,221) (23,264,265)
(24,143,145) (25,312,313) (27,364,365) (28,45,53)
(28,195,197) (29,420,421) (31,480,481) (32,255,257)
(33,56,65) (33,544,545) (35,612,613) (36,77,85)
(36,323,325) (37,684,685)
And on and on...
Remember that if then 
so I did not include 6,8,10 or 10,24,26.
Special Right triangles
45-90-45
Theorem
Say you have a right triangle with angles 45, 45 and 90. Then if (leg) is 
, then (long side) is 
Proof
If this triangle has two equal angles then it has two equal sides. Therefore we can make an equation: 
. 
means side and is the hypotenuse. 
. We proved it!
