Pythagorean Theory

Pythagoras Pythagoras lived in the 500's BC, and was one of the first Greek mathematical thinkers.

The Egyptians knew that a triangle with sides 3, 4, and 5 make a 90o angle. As a matter of fact, they had a rope with 12 evenly spaced knots like this one:                                                                               that they used to build perfect corners in their buildings and pyramids. It is believed that they only knew about the 3, 4, 5 triangle and not the general theorem that applies to all right triangles.

So why is it called the Pythagorean Theorem? Even though the theorem was known long before his time, Pythagoras generalized it and made it popular. It was Pythagoras who is attributed with its first geometrical demonstration.

The Theorem (In simple terms) When the two shorter sides in a right triangle are squared and then added, the sum equals the square of the longest side or hypotenuse.

Theorem Proof a2 + b2 = h2 32 + 42 = 52 9 + 16 = 25 52 = 25 42 = 16 http://www.pbs.org/wgbh/nova/proof/puzzle/images/example.gif 32 = 9 a2 + b2 = h2 32 + 42 = 52 9 + 16 = 25

Finding the Unknown Side 6 cm 8 cm h First, sketch the diagram into your notebook.

a2 + b2 = h2 62 + 82 = h2 Then copy out the formula. Identify the hypotenuse: “c” in the formula. Always the side opposite the right angle 6 cm 8 cm h Note: it doesn’t matter which of the two sides is a or b. Make the substitutions into the formula. 62 + 82 = h2

BEDMAS tells us to do the exponents first. Do the calculations BEDMAS tells us to do the exponents first. 62 + 82 = h2 6 cm 8 cm h 36 + 64 = h2 Then add. To find the value of c: You have to do the opposite function of squaring a number. The opposite function is √ (square root). 100 = h2 10 = h

Solving if one of the sides is missing. Always start by writing the formula. a2 + b2 = h2 C is always the hypotenuse The side opposite the right angle Make the substitutions. a2 + 92 = 152 Calculate and rewrite 15 cm a2 + 81 = 225 -81 Isolate the variable Get rid of the +81! 9 cm -81 Cancel Rewrite the equation a a2 = 144 To isolate the variable, do the opposite function. Take the square root of each side.

Using the Pythagorean Theorem, solve for the missing side. 11 cm 26 cm

Solve for ϰ 28 cm 23 cm Check your understanding

Practical Applications The Pythagorean Theorem is used in everyday life all the time. Carpenters use it to check to see if a structure is square. If you know two sides of any right angle triangle, you can calculate the length of the unknown side.

Baseball A baseball diamond consists of four bases each 90 ft apart situated at 90º angles from each other. If the catcher tries to throw out a runner at second base, how far does he throw the ball? 90ft

Around the house You have to fix a window that is 12 ft high from the ground. You have a bush at the bottom of the window, so you will need to place the ladder back 3 ft from the base of the house. How long will the ladder need to be? Note: Ladders can only be bought in full feet, not decimals.

3, 4, 5 Rule Carpenters use the “3, 4, 5 Rule” to make sure that a deck is square. They measure a spot on one side and mark the 3 ft line. They measure a spot on the other side and mark the 4 ft line. If the measurement between the two marks equals 5 ft, they know the deck is “square”. Prove this by using the Pythagorean Theorem. x x

Now Some more Practical applications Practical Questions