Pythagorean TheoremPythagorean TheoremM8G2. Students will understand and use the Pythagorean M8G2. Students will understand and use the Pythagorean theorem. theorem. a. Apply properties of right triangles, including the Pythagorean a. Apply properties of right triangles, including the Pythagorean theorem. theorem. b. Recognize and interpret the Pythagorean theorem as a b. Recognize and interpret the Pythagorean theorem as a statement about areas of squares on the sides of a right statement about areas of squares on the sides of a right triangle. 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 HistoryA Little Bit of History

Over 2,500 years Over 2,500 years ago, a Greek ago, a Greek mathematician named mathematician named Pythagoras Pythagoras developed a proof developed a proof that the relationship that the relationship between the between the hypotenuse and the hypotenuse and the legs is true for legs is true for all all right triangles.right triangles.

This relationship can This relationship can be stated as:be stated as:

and is known as the and is known as the Pythagorean Theorem.Pythagorean Theorem.

Who Was Pythagoras?Who Was Pythagoras?

Greek Mathematician and Greek Mathematician and philosopherphilosopher

582 BC-496 BC582 BC-496 BC Part of a secret society Part of a secret society

call Pythgoreans. call Pythgoreans. Believed that whole Believed that whole

numbers and their ratios numbers and their ratios could account for could account for Geometrical properties.Geometrical properties.

They were disturbed by They were disturbed by the discovery of irrational the discovery of irrational numbers!numbers!

Pythagorean TheoremPythagorean Theorem

In any right triangle, In any right triangle, the square of the the square of the length of the length of the hypotenuse is equal hypotenuse is equal to the sum of the to the sum of the squares of the lengths squares of the lengths of the legs. of the legs.

   

a, b are legs.a, b are legs.    c is the hypotenuse    c is the hypotenuse(c is across from the (c is across from the right angle). right angle).

Pythagorean Triples Pythagorean Triples

There are certain sets of There are certain sets of numbers that have a very numbers that have a very special property.  special property. 

Not only do these Not only do these numbers satisfy the numbers satisfy the Pythagorean Theorem, Pythagorean Theorem, but any multiples of these but any multiples of these numbers also satisfy the numbers also satisfy the Pythagorean Theorem.  Pythagorean Theorem.  

The most common The most common Pythagorean Triples Pythagorean Triples are: are:

3, 4, 5 3, 4, 5

5, 12, 13 5, 12, 13

8, 15, 178, 15, 17

REMEMBER:REMEMBER:

The Pythagorean Theorem The Pythagorean Theorem ONLYONLY works in works in Right TrianglesRight Triangles!!

Example 1:Example 1:

Find x.Find x.

Answer: 10 mAnswer: 10 m

This problem could This problem could also be solved using also be solved using the Pythagorean the Pythagorean Triple 3, 4, 5.  Since 6 Triple 3, 4, 5.  Since 6 is 2 times 3, and 8 is is 2 times 3, and 8 is 2 times 4, then x must 2 times 4, then x must be 2 times 5. be 2 times 5.

Example 2:Example 2:

A triangle has sides A triangle has sides 6, 7 and 10. 6, 7 and 10. Is it a right triangle?Is it a right triangle?

Since the Since the Pythagorean Pythagorean Theorem is NOT Theorem is NOT true, this triangle is true, this triangle is NOT a right triangle.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 3Example 3 A ramp was constructed to load a truck.  If the A ramp was constructed to load a truck.  If the

ramp is 9 feet long and the horizontal distance ramp is 9 feet long and the horizontal distance from the bottom of the ramp to the truck is 7 feet, from the bottom of the ramp to the truck is 7 feet, what is the vertical height of the ramp?what is the vertical height of the ramp?

The height of the ramp is 5.7 feet.

Problem #1Problem #1To get from point A to point B you must avoid To get from point A to point B you must avoid

walking through a pond.  To avoid the pond, walking through a pond.  To avoid the pond, you must walk 34 meters south and 41 meters you must walk 34 meters south and 41 meters east.  To the east.  To the nearestnearest metermeter, how many meters , how many meters would be saved if it were possible to walk would be saved if it were possible to walk through the pond?  through the pond? 

A. 22 B. 34C. 53 D. 75 

Problem #2Problem #2

A baseball diamond is a square with sides of A baseball diamond is a square with sides of 90 feet.  What is the shortest distance, to 90 feet.  What is the shortest distance, to the the nearest tenth nearest tenth of a foot, between first of a foot, between first base and third base? base and third base?

A. 90.0B. 127.3 C. 180.0 D. 180.7

Problem #3Problem #3

A suitcase measures 24 inches long and 18 A suitcase measures 24 inches long and 18 inches high.  What is the diagonal length of inches high.  What is the diagonal length of the suitcase to the the suitcase to the nearest tenthnearest tenth of a foot? of a foot?

A. 2.5 B. 2.9 C. 26.5 D. 30.0

Problem #4Problem #4

In a computer catalog, a computer monitor is In a computer catalog, a computer monitor is listed as being 19 inches.  This distance is listed as being 19 inches.  This distance is the diagonal distance across the screen.  the diagonal distance across the screen.  If the screen measures 10 inches in If the screen measures 10 inches in height, what is the actual width of the height, what is the actual width of the screen to the screen to the nearest inchnearest inch?  ? 

A. 10 B. 14 C. 16 D. 19

Problem #5Problem #5

The older floppy diskettes measured 5 and The older floppy diskettes measured 5 and 1/4 inches on each side.  What was the 1/4 inches on each side.  What was the diagonal length of the diskette to the diagonal length of the diskette to the nearest tenth nearest tenth of an inch? of an inch?

A. 5.3 B. 6.5 C. 7.4 D. 7.6

Problem #6Problem #6

Ms. Green tells you that a right triangle has Ms. Green tells you that a right triangle has a hypotenuse of 13 and a leg of 5.  She a hypotenuse of 13 and a leg of 5.  She asks you to find the other leg of the asks you to find the other leg of the triangle without using paper and pencil.  triangle without using paper and pencil.  What is your answer? What is your answer?

A. 5 B. 8 C. 10 D. 12

Problem #7Problem #7

Two joggers run 8 miles north and then 5 Two joggers run 8 miles north and then 5 miles west.  What is the shortest distance, miles west.  What is the shortest distance, to the to the nearest tenthnearest tenth of a mile, they must of a mile, they must travel to return to their starting point? travel to return to their starting point?

A. 8.4 B. 9.5 C. 9.4 D. 13.1

Problem #8Problem #8

Find x .Find x .

A. 6 B. 8 C. 10 D. 12

Problem #9Problem #9

Oscar's dog house is shaped like a tent.  The Oscar's dog house is shaped like a tent.  The slanted sides are both 5 feet long and the slanted sides are both 5 feet long and the bottom of the house is 6 feet across.  What bottom of the house is 6 feet across.  What is the height of his dog house, in feet, at its is the height of his dog house, in feet, at its tallest point? tallest point?

A. 3 B. 4 C. 4.5 D. 5

Problem #10Problem #10

Seth made a small quadrilateral table for his Seth made a small quadrilateral table for his workroom.  The sides of the table are 36" workroom.  The sides of the table are 36" and 18".  If the diagonal of the table and 18".  If the diagonal of the table measures 43", is the table rectangular?  A measures 43", is the table rectangular?  A table which is “rectangular" has right table which is “rectangular" has right angles at the corners. angles at the corners.

A. Yes B. No

Did You Know?Did You Know?

One of our American Presidents discovered a One of our American Presidents discovered a proof for the Pythagorean Theorem before he was proof for the Pythagorean Theorem before he was elected President!elected President!

There are at least 367 proofs of the Pythagorean There are at least 367 proofs of the Pythagorean Theorem.Theorem.

GPS only suggests using the proof of “areas of GPS only suggests using the proof of “areas of squares on the sides of a right triangle” M8G2: b.squares on the sides of a right triangle” M8G2: b.

President Garfield’s ProofPresident Garfield’s Proof

This figure was used by This figure was used by President Garfield in President Garfield in proving the Pythagorean proving the Pythagorean theorem. theorem.

His method is based on His method is based on the fact that the area of the fact that the area of the trapezoid ACED is the trapezoid ACED is equal to the sum of the equal to the sum of the areas of the three right areas of the three right triangles ACB, ABD, and triangles ACB, ABD, and BED. BED.

Animated ProofAnimated Proof

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

ReferencesReferences

Animated Proof of Pythagorean Theorem Animated Proof of Pythagorean Theorem http://www.nadn.navy.mil/MathDept/mdm/http://www.nadn.navy.mil/MathDept/mdm/pyth.htmlpyth.html

Regents Prep Regents Prep http://www.regentsprep.org/Regents/math/http://www.regentsprep.org/Regents/math/math-topic.cfm?TopicCode=fpythmath-topic.cfm?TopicCode=fpyth

PBS Proving the Pythagorean Theorem PBS Proving the Pythagorean Theorem http://www.pbs.org/teachers/mathline/conchttp://www.pbs.org/teachers/mathline/concepts/historyandmathematics/activity1.shtmepts/historyandmathematics/activity1.shtm