Right triangles and the Pythagorean theorem
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RIGHT TRIANGLES AND THE PYTHAGOREAN THEOREM Learning about Click to begin… Joe Deevy
description
Right triangles and the Pythagorean theorem. Learning about. Click to begin…. Joe Deevy. Menu. Right triangle basics. Pythagoras and his theorem. Using the Pythagorean theorem. Proving the Pythagorean theorem. Credits & Standards. Joe Deevy. What’s a right triangle?. - PowerPoint PPT Presentation
Transcript of Right triangles and the Pythagorean theorem
RIGHT TRIANGLES AND THE PYTHAGOREAN THEOREM
Learning about
Click to begin…
Joe Deevy
MENU
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 mmb) 10 mmc) 3.74 mmd) 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?
Joe Deevy
NoVote 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.
