2. Who was Pythagoras?
An ancient Greek thinker who was fond of poetry
and literature
He himself was not a geometer
His followers, dubbed the Pythagorean
Brotherhood, were a religious cult
Contrary to popular belief, Pythagoras was NOT
the founder of the Pythagorean Theorem
The Pythagorean Brotherhood founded the theorem
over 100 years after Pythagoras had died
3. What It Relates to:
Any and all right triangles
This means that this angle is a right angle
A right angle = 90º
Since there are 180º in a triangle, the other 2 angles must add
up to 90º
4. What Is It?
Let’s label the triangle by its angles ABC
If the triangle’s angles are ABC, then the sides
opposite those angles are a, b, and c,
respectively
AC
B
b
a
c
5. What Is It?
a and b are the sides, or legs, of the right
triangle
c is the hypotenuse, or the side opposite the
right angle, which is always the longest side of
the right triangle
In any triangle, the sums of any 2 sides is
greater than the length of the 3rd
side, so:
a + b > c
a + c > b
b + c > a
6. What Is It?
However, sometimes when we square the sides,
the sum of the squares is equal to the square of
the hypotenuse
a2
+ b2
= c2
This is the Pythagorean Theorem
The Pythagorean Theorem states:
In any right triangle, the sum of the squares of the 2 sides is
equal to the square of the hypotenuse.
Conversely, if the sum of the squares of the 2 sides is equal
to the square of the hypotenuse, then you have a right
triangle.
7. What Is It?
What does this mean?
It means that, if you have a right triangle, you
know that the squares of the 2 sides will
always add up to the square of the
hypotenuse.
It means that if you have a triangle in which
the squares of the 2 sides add up to the
square of the hypotenuse, you know you have
a right triangle.
8. What Is It?
Let’s look at some proofs:
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
1 2 3
4 5 6
7 8 9
9. What Is It?
If you add up all of the little squares, i.e. 9
+ 16, you get 25. This is in accord with 32
+ 42
= 52
Here, we see that the lengths of the sides
are 3 and 4, and the length of the
hypotenuse is 5.
Using the Pythagorean Theorem, we get
32
+ 42
= 52
, or 9 + 16 = 25.
11. What Is It?
What is the area of the large blue square?
By multiplying the length by the width, we get (a + b)2
Simplify to a2
+ 2ab + b2
By adding up the areas of 4 small blue triangles and
the 1 small purple square, we get, 4(1/2)ab + c2
Simplify to 2ab + c2
If we set the areas equal to each other, we get a2
+
2ab + b2
= 2ab + c2
The 2ab’s cancel out, so we get a2
+ b2
= c2
Thus, the Pythagorean Theorem has been once
again proven.
12. What Is It?
The Pythagorean Theorem specifically refers to
squares of the sides, however any 2-
dimensional relation between the 2 sides and
the hypotenuse would work
If we drew circles on each side and the hypotenuse,
with the sides as the diameter of each respective
circle, then the areas of the circles on the 2 sides will
sum up to the area of the circle on the hypotenuse
Algebraically, this is seen as ka2
+ kb2
= kc2
since k
can be factored out, where k is some constant
13. Practice
Find the missing lengths:
a = 3, b = 4, c = ?
a = 7, b = ?, c = 25
a = ?, b = 12, c = 13
Are triangles with the following lengths right
triangles?
a = 7, b = 8, c = 9
a = 12, b = 16, c = 20
a = 11, b = 58, c = 61
14. Practice
To get from point A to point B you must avoid
walking through a pond. To avoid the pond, you
must walk 34 meters south and 41 meters east.
To the nearest meter, how many meters would
be saved if it were possible to walk through the
pond?
A baseball diamond is a square with sides of 90
feet. What is the shortest distance, to
thenearest tenth of a foot, between first base
and third base?
15. Practice
In a computer catalog, a computer monitor is
listed as being 19 inches. This distance is the
diagonal distance across the screen. If the
screen measures 10 inches in height, what is the
actual width of the screen to the nearest inch?
Oscar's dog house is shaped like a tent. The
slanted sides are both 5 feet long and the
bottom of the house is 6 feet across. What is
the height of his dog house, in feet, at its tallest
point?
16. Why Does It Matter?
The Pythagorean Theorem allows us to do many
things in real-life situations.
The professional fields in which it is useful are:
Civil Engineering
Construction
Astronomy
Physics
Particle Physics
Advanced Mathematics
Ancient Warfare
17. Why Does It Matter?
Civil Engineering:
Building bridges
Measuring distances across rivers in order to
determine the lengths of proposed bridges
Building foundations of skyscrapers
Construction
Measuring angles
Ensuring solid and level foundations
18. Why Does It Matter?
Astronomy
Measuring distances in a 3-dimensional
space
Calculating shadows cast by astronomical
bodies
Physics
Determining pressure in bridge construction
Understanding ramps, levers, and screws
19. Why Does It Matter?
Particle Physics
Calculating distances of particles in 3-dimensional space
Advanced Mathematics
Pythagorean Triples
Trigonometry and the Unit Circle
Vectors
Calculating distances between points on a Cartesian Plane
Ancient Warfare
The Ancient Romans used it to measure the distance that
catapults had to be from their target
20. Food for Thought
The distance formula for 2 points on a Cartesian Plane is derived
from the Pythagorean Theorem
The distance formula is d = √[(x2 – x1)2
+ (y2 – y1)2
]
This is simply a variation on c = √(a2
+ b2
), which is the Pythagorean
Theorem if you solve for c2
Pythagorean Triples are sets of 3 numbers that fit the criteria of a2
+
b2
= c2
Since any set of Pythagorean Triples can be multiplied by an infinite
amount of constants, there are an infinite amount of Pythagorean
Triples
If triangles with side lengths that corresponded to every Primitive
Pythagorean Triple (reduced by greatest common factor) were drawn on
a Cartesian Plane, we would end up with a unit circle, which is where
our Trigonometric functions come from
