Right triangles and the Pythagorean theorem

Right triangles and the Pythagorean theorem Learning about Click to begin… Joe Deevy

Menu Right triangle basics Pythagoras and his theorem Using the Pythagoreantheorem Proving the Pythagoreantheorem Credits & Standards Joe Deevy

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 side opposite the right angle is thehypotenuse The sides touching the right angle are legs

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? Correct! The hypotenuse is opposite the right angle! Great job!

Can you name the sides? No, that’s a leg – it touches the right angle

Who was Pythagoras? …and what did he do? • 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 Joe Deevy

What’s the Pythagorean theorem? Name the legs of a right triangle a and b. Name the hypotenuse c. c b a Pythagorean Theorem: The lengths of sides a, b, and c are related by the equation: a2 + b2 = c2

What does it mean? a2 + b2 = c2 c = ? b = 3 inch a =4 inch Look at this example triangle. The lengths of legsa and b are given. We can use the Pythagorean theorem to find the length of the hypotenusec.

What does it mean? a2 = 42 = 16 b2 = 32 = 9 c = 5 inch c = ? b = 3 inch a2 + b2 = c2 16+9=c2 a =4 inch 25 = c2 5 = c (take square route of 25) Click arrow to see the next step… Finished!

Now You Try It… b = 6 mm What is the length of the hypotenuse c?Click the correct answer: a) 14 mm • a = 8 mm b) 10 mm c = ? c) 3.74 mm d) 100 mm

Now You Try It… b = 6 mm Correct! c = 10 mm • a = 8mm c = ? Great job!

Now You Try It… b = 6 mm No, the correct answer is b) 10 mm Here’s how you find c: • a = 8 mm a2 = 64 b2 = 36 a2 + b2 = c2 64 + 36 = c2 100 = c2 c = 10 mm c = ?

How did he prove it? Do you remember how to find the area of a square? Area = side2 side = b Area = b2 • side = a • Area= a2 side = c Area = c2 Joe Deevy

How did he prove it? Put these squares around a right triangle a2 + b2 = c2 means that the blue and green squares together cover the same area as the area of the red square by itself! Area = c2 Area = b2 c b • a • Area= a2 How would you prove that? Joe Deevy

Here’s one way to show it… Click on the movie camera, to see a video: Joe Deevy

But is it really a Proof? Vote by clicking Yes or No: Yes No Joe Deevy

You’re right! it’s not a proof! • 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! Joe Deevy

Actually, it’s not a proof • 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… Joe Deevy

A real Proof of the Pythagorean theorem Click on the movie camera, to see a video proof: Joe Deevy

An Interactive Proof of the Pythagorean theorem 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 Joe Deevy

The END! Congratulations! Now you know what the Pythagorean theorem is all about! Joe Deevy

credits 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 Joe Deevy

standards 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. Joe Deevy

Right triangles and the Pythagorean theorem