RIGHT TRIANGLES AND THE PYTHAGOREAN THEOREM

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Joe Deevy

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Joe Deevy

RIGHT TRIANGLE BASICS

PYTHAGORAS AND HIS THEOREM

USING THE PYTHAGOREAN THEOREMPROVING THE PYTHAGOREAN THEOREMCREDITS & STANDARDS

WHAT’S A RIGHT TRIANGLE?

It’s a triangle with one 90° angle

The 90° angle usually has a square drawn inside

WHAT ARE THE PARTS OF A RIGHT TRIANGLE?The triangle’s sides have names:

The sides touching the right angle are legs

WHAT ARE THE PARTS OF A RIGHT TRIANGLE?The triangle’s sides have names:

The sides touching the right angle are legs

The side opposite the right angle is the hypotenuse

WHAT DID YOU CALL THAT?

hypotenuse (hi-POT-ten-noose)

CAN YOU NAME THE SIDES?

click on one of the legs:

CAN YOU NAME THE SIDES? Correct! The legs touch the right angle!

Great job!

CAN YOU NAME THE SIDES? No, that’s the hypotenuse – it’s opposite the right angle

Try again!

CAN YOU NAME THE SIDES?

click on the hypotenuse:

CAN YOU NAME THE SIDES?

Great job!

Correct! The hypotenuse is opposite the right angle!

CAN YOU NAME THE SIDES? No, that’s a leg – it touches the right angle

WHO WAS PYTHAGORAS?…and what did he do?

Joe Deevy

Pythagoras lived about 2500 years ago. He lived most of his life in what is now Sicily and southern Italy.

He was an early Greek thinker, and was interested in music, philosophy, and mathematics. Pythagoras had men and women who followed him to learn from him.

2000 years before Pythagoras, people in Samaria found an important rule about right triangles. However, they could not prove that the rule was always true. Pythagoras proved it, so the rule became known as the “Pythagorean Theorem” in his honor.

Pythagoras teaching

WHAT’S THE PYTHAGOREAN THEOREM?Name the legs of a right triangle a and b. Name the hypotenuse c.

Pythagorean Theorem:The lengths of sides a, b, and c are related by the equation:

a

bc

a2 + b2 = c2

WHAT DOES IT MEAN?

Look at this example triangle. The lengths of legs a and b are given. We can use the Pythagorean theorem to find the length of the hypotenuse c.

a = 4 inch

b = 3 inchc = ?

a2 + b2 = c2

c = 5 inch

WHAT DOES IT MEAN?

5 = c(take square route of 25)

a2 = 42 = 16

b2 = 32 = 9a2 + b2 = c2

16 + 9 = c2

25 = c2a = 4 inch

b = 3 inchc = ?

Finished!Click arrow to see the next step…

NOW YOU TRY IT…

a = 8 mm

b = 6 mm

c = ?

What is the length of the hypotenuse c?Click the correct answer:a) 14 mm

b) 10 mm

c) 3.74 mm

d) 100 mm

NOW YOU TRY IT…

a = 8 mm

b = 6 mm

c = ?

Correct! c = 10 mm

Great job!

NOW YOU TRY IT…

a = 8 mm

b = 6 mm

c = ?

No, the correct answer is b) 10 mmHere’s how you find c:a2 = 64b2 = 36a2 + b2 = c2

64 + 36 = c2

100 = c2

c = 10 mm

HOW DID HE PROVE IT?

Joe Deevy

Do you remember how to find the area of a square?

Area = side2

side = a

Area = a2

side = bArea = b2

side = c Area = c2

HOW DID HE PROVE IT?

Joe Deevy

Put these squares around a right triangle

a b c

Area = c2

Area = a2

Area = b2

a2 + b2 = c2 means that the blue and green squares together cover the same area as the area of the red square by itself!

How would you prove that?

HERE’S ONE WAY TO SHOW IT…

Joe Deevy

Click on the movie camera, to see a video:

BUT IS IT REALLY A PROOF?

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No

Vote by clicking Yes or No:

Yes

YOU’RE RIGHT! IT’S NOT A PROOF!

Joe Deevy

The water demonstration gives the idea, but:• it’s not exact• it doesn’t show that the theorem always works

Let’s look at a real proof… Congratulations!

ACTUALLY, IT’S NOT A PROOF

Joe Deevy

The water demonstration gives the idea, but:• it’s not exact• it doesn’t show that the theorem always works

Let’s look at a real proof…

A REAL PROOF OF THE PYTHAGOREAN THEOREM

Joe Deevy

Click on the movie camera, to see a video proof:

AN INTERACTIVE PROOF OF THE PYTHAGOREAN THEOREM

Joe Deevy

Here are two proof demos that you can play with:

Proof 1

Proof 2

Hint: For this demo, move the pink circle if you like, then use the slider at the bottom of the window to see the proof.

Hint: Use the “Start” and “Next” buttons to step through the proof

THE END!

Joe Deevy

Congratulations! Now you know what the Pythagorean theorem is all about!

CREDITS

Joe Deevy

A. Bogomolny, Pythagorean Theorem By Hinged Dissection 3 (Third Interactive Variant) from Interactive Mathematics Miscellany and Puzzles.http://www.cut-the-knot.org/Curriculum/Geometry/HingedPythagoras3.shtml. Accessed 17 July 2010

Math Cove - Teaching and Learning Mathematics with Java.http://oneweb.utc.edu/~Christopher-Mawata/geom/geom7.htm . Accessed 17 July 2010

YouTube: Teorema de Pitágoras.http://www.youtube.com/watch?v=hTxqdyGjtsA&feature=related . Accessed 17 July 2010

YouTube: The Pythagorean Theorem animation. http://

www.youtube.com/watch?v=7-BU5y6jZpw . Accessed 17 July 2010

STANDARDS

Joe Deevy

PDE:

2.3.G.C: Use properties of geometric figures and measurement formulas to solve for a missing quantity.

2.9.G.A: Identify and use properties and relations of geometric figures; create justifications for arguments related to geometric relations.

ISTE NETS:

4d.   Students use multiple processes and diverse perspectives to explore alternative solutions.

5b.   Students exhibit a positive attitude toward using technology that supports collaboration, learning, and productivity.

6d.   Students transfer current knowledge to learning of new technologies.