Pythagorean theorem

a, b are legs. c is the hypotenuse (across from the right angle). Pythagorean Theorem The sum of the squares of the sides of a right triangle is equal to the square of the hypotentuse. We can represent this relationship using a formula… 𝑎 2 + 𝑏 2 = 𝑐 2 a, b are legs. c is the hypotenuse (across from the right angle).

Animated Proof of the Pythagorean Theorem Below is an animated proof of the Pythagorean Theorem. Starting with a right triangle and squares on each side, the middle size square is cut into congruent quadrilaterals (the cuts through the center and parallel to the sides of the biggest square). Then the quadrilaterals are hinged and rotated and shifted to the big square. Finally the smallest square is translated to cover the remaining middle part of the biggest square. A perfect fit! Thus the sum of the squares on the smaller two sides equals the square on the biggest side. Afterward, the small square is translated back and the four quadrilaterals are directly translated back to their original position. The process is repeated forever.

Second Animated Proof of the Pythagorean Theorem

Who would even think to do that? Pythagoras (~580-500 B.C.) He was a Greek philosopher responsible for important developments in mathematics, astronomy and the theory of music.

How does this help me in Reality… David Swasey: He is an Electronic Material Officer in the United States Navy. “Oh! I use geometry every day”. Dave at his job 

Other careers that use Pythagorean Theorem -Computer and information systems managers  -Construction managers  -Engineering and natural sciences managers  -Professional and Related  -Computer software engineers  -Mathematicians  -Architects, except landscape and naval  -Landscape architects  -Surveyors, cartographers, photogrammetrists, and surveying technicians  -Aerospace engineers  -Chemical engineers  -Civil engineers  -Computer hardware engineers  -Electrical and electronics engineers, except computer  -Environmental engineers  -Industrial engineers, including health and safety  -Materials engineers  -Mechanical engineers  -Nuclear engineers  -Drafters  -Biological scientists  -Conservation scientists and foresters  -Atmospheric scientists  -Chemists and materials scientists  -Environmental scientists and geoscientists  -Physicists and astronomers  -Lawyers  -Archivists, curators, and museum technicians  -Writers and editors  -Optometrists  -Physicians and surgeons  Veterinarians  -Opticians, dispensing  -Farming and Related  -Agricultural workers

Ok so what was the point? The point of using Pythagorean theorem is to find missing sides of Right Triangles. We can identify the parts of a right triangle, and use them in Pythagorean Theorem!

Using a2 + b2 = c2 x 15 20 Looking for length of the hypotenuse 𝑎2 + 𝑏2 = 𝑐2 Pythagorean Theorem Looking for length of the hypotenuse 152 + 202 = 𝑐2 Substitution 225 + 400 = 𝑐2 Simplify x 15 625 = 𝑐2 Simplify 625 = 𝑐 2 20 Square root both sides 25=𝑐 Simplify

Using a2 + b2 = c2 10 6 x Looking for length of a Leg! 𝑎2 + 𝑏2 = 𝑐2 Pythagorean Theorem Looking for length of a Leg! 62 +𝑏2 =102 Substitution 36 + 𝑏 2 =100 Simplify 10 6 𝑏 2 =64 Subtraction 𝑏 2 = 64 x Square root both sides 𝑏=8 Simplify

You Try! 3 13 8 x x 5 a2 + b2 = c2 a2 + b2 = c2 82 + 32 = c2

Converse of Pythagorean theorem If 𝑎2 + 𝑏2 = 𝑐2, then the triangle is a right triangle If 𝑎2 + 𝑏2> 𝑐2 then the triangle is ACUTE If 𝑎2 + 𝑏2< 𝑐2 then the triangle is OBTUSE In english: If we know that the leg squared, plus the other leg squared, is equal to the hypotenuse squared…Then the triangle must be a RIGHT TRIANGLE!

Example: Are these the side lengths of a right triangle? 𝑎2 + 𝑏2 = 𝑐2 Pythagorean Theorem 3 32 +42 =52 Substitution 9 +16 =25 Simplify 4 25 =25 5 Simplify YES! This is a right triangle 

You Try!: 9 9 7 6 16 7 Are these the side lengths of a right triangle? NO! Not a triangle! 16 7 NO! This is a not a right triangle! It is ACUTE!!

Pythagorean Triples Pythagorean triples are three positive integers such that 𝑎 2 + 𝑏 2 = 𝑐 2 . There are a few that are easy to remember, like the following: 3,4,5 5,12,13 7,24,25 To determine if three sets of numbers form a triple, we plug them into 𝑎 2 + 𝑏 2 = 𝑐 2 and see if we reach equality. For example: 3 2 + 4 2 = 5 2 9+16=25 25=25