Pythagorean Theorem M8G2. Students will understand and use the Pythagorean theorem. a. Apply properties of right triangles, including the Pythagorean theorem. b. Recognize and interpret the Pythagorean theorem as a statement about areas of squares on the sides of a right triangle. Related Standards from GADOE Framework: M8P2. Students will reason and evaluate mathematical arguments. a. Recognize reasoning and proof as fundamental aspects of mathematics. b. Make and investigate mathematical conjectures. c. Develop and evaluate mathematical arguments and proofs. d. Select and use various types of reasoning and methods of proof.

A Little Bit of History Over 2,500 years ago, a Greek mathematician named Pythagoras developed a proof that the relationship between the hypotenuse and the legs is true for all right triangles. This relationship can be stated as: and is known as the Pythagorean Theorem.

Who Was Pythagoras? Greek Mathematician and philosopher 582 BC-496 BC Part of a secret society call Pythgoreans. Believed that whole numbers and their ratios could account for Geometrical properties. They were disturbed by the discovery of irrational numbers!

Pythagorean Theorem In any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.   a, b are legs.     c is the hypotenuse (c is across from the right angle).

Pythagorean Triples The most common Pythagorean Triples are: 3, 4, 5 There are certain sets of numbers that have a very special property.  Not only do these numbers satisfy the Pythagorean Theorem, but any multiples of these numbers also satisfy the Pythagorean Theorem.   The most common Pythagorean Triples are: 3, 4, 5 5, 12, 13 8, 15, 17

REMEMBER: The Pythagorean Theorem ONLY works in Right Triangles!

Example 1: Find x. Answer: 10 m This problem could also be solved using the Pythagorean Triple 3, 4, 5.  Since 6 is 2 times 3, and 8 is 2 times 4, then x must be 2 times 5.

Example 2: A triangle has sides 6, 7 and 10. Is it a right triangle? Since the Pythagorean Theorem is NOT true, this triangle is NOT a right triangle. Let a = 6, b = 7 and c = 10.  The longest side MUST be the hypotenuse, so c = 10.  Now, check to see if the Pythagorean Theorem is true.

Example 3 A ramp was constructed to load a truck.  If the ramp is 9 feet long and the horizontal distance from the bottom of the ramp to the truck is 7 feet, what is the vertical height of the ramp? The height of the ramp is 5.7 feet.

Problem #1 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. 22 B. 34 C. 53 D. 75 

Problem #2 A baseball diamond is a square with sides of 90 feet.  What is the shortest distance, to the nearest tenth of a foot, between first base and third base? A. 90.0 B. 127.3 C. 180.0 D. 180.7

Problem #3 A suitcase measures 24 inches long and 18 inches high.  What is the diagonal length of the suitcase to the nearest tenth of a foot? A. 2.5 B. 2.9 C. 26.5 D. 30.0

Problem #4 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?  A. 10 B. 14 C. 16 D. 19

Problem #5 The older floppy diskettes measured 5 and 1/4 inches on each side.  What was the diagonal length of the diskette to the nearest tenth of an inch? A. 5.3 B. 6.5 C. 7.4 D. 7.6

Problem #6 Ms. Green tells you that a right triangle has a hypotenuse of 13 and a leg of 5.  She asks you to find the other leg of the triangle without using paper and pencil.  What is your answer? A. 5 B. 8 C. 10 D. 12

Problem #7 Two joggers run 8 miles north and then 5 miles west.  What is the shortest distance, to the nearest tenth of a mile, they must travel to return to their starting point? A. 8.4 B. 9.5 C. 9.4 D. 13.1

Problem #8 Find x . A. 6 B. 8 C. 10 D. 12

Problem #9 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? A. 3 B. 4 C. 4.5 D. 5

Problem #10 Seth made a small quadrilateral table for his workroom.  The sides of the table are 36" and 18".  If the diagonal of the table measures 43", is the table rectangular?  A table which is “rectangular" has right angles at the corners. A. Yes B. No

Did You Know? One of our American Presidents discovered a proof for the Pythagorean Theorem before he was elected President! There are at least 367 proofs of the Pythagorean Theorem. GPS only suggests using the proof of “areas of squares on the sides of a right triangle” M8G2: b.

President Garfield’s Proof This figure was used by President Garfield in proving the Pythagorean theorem. His method is based on the fact that the area of the trapezoid ACED is equal to the sum of the areas of the three right triangles ACB, ABD, and BED.

Animated Proof http://www.nadn.navy.mil/MathDept/mdm/pyth.html

References Animated Proof of Pythagorean Theorem http://www.nadn.navy.mil/MathDept/mdm/pyth.html Regents Prep http://www.regentsprep.org/Regents/math/math-topic.cfm?TopicCode=fpyth PBS Proving the Pythagorean Theorem http://www.pbs.org/teachers/mathline/concepts/historyandmathematics/activity1.shtm