The Pythagorean Theorem

By Kaamil Ali

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

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

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

ac

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

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.

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.

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

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.

What Is It? Another proof:

cc

c c

b

b

b

b

a

a

a

a

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.

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

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

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?

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?

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

Why Does It Matter?

Civil Engineering:Building bridgesMeasuring distances across rivers in order to

determine the lengths of proposed bridgesBuilding foundations of skyscrapers

ConstructionMeasuring anglesEnsuring solid and level foundations

Why Does It Matter?

AstronomyMeasuring distances in a 3-dimensional

spaceCalculating shadows cast by astronomical

bodies Physics

Determining pressure in bridge constructionUnderstanding ramps, levers, and screws

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

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