Pythagorean Theorem by Bobby Stecher

The Pythagorean Theorem as some students see it. a c b a2+b2=c2a2+b2=c2

a2+b2=c2a2+b2=c2 A better way a2a2 b2b2 c2c2 c b a

Pythagorean Triples (3,4,5) (5,12,13) (7,24,25) (8,15,17) (9,40,41) (11,60,61) (12,35,37) (13,84,85) (16,63,65)

Pythagorean Triples

The distance formula. (x 1,y 1 ) (x 2,y 2 )a = x 2 -x 1 b = y 2 -y 1 c = distance The Pythagorean Theorem is often easier for students to learn than the distance formula.

Proof of the Pythagorean Theorem from Euclid Euclid’s Proposition I.47 from Euclid’s Elements.

Proof of the Pythagorean Theorem Line segment CN is perpendicular to AB and segment CM is an altitude of ΔABC.

Proof of the Pythagorean Theorem Triangle ΔAHB has base AH and height AC. Area of the triangle ΔAHB is half of the area of the square with the sides AH and AC.

Proof of the Pythagorean Theorem Triangle ΔACG has base AG and height AM. Area of the triangle ΔACG is half of the area of the rectangle AMNG.

Proof of the Pythagorean Theorem AG is equal to AB because both are sides of the same square. ΔACG is equal ΔAHB by SAS. AC is equal to AH because both are sides of the same square. Angle <CAG is equal to <HAB. Both angles are formed by adding the angle <CAB to a right angle. Recall that ΔACG is half of rectangle AMNG and ΔAHB is equal to half of square ACKH. Thus square ACKH is equal to rectangle AMNG.

Proof of the Pythagorean Theorem Triangle Δ MBE has base BE and height BC Triangle Δ MBE is equal to half the area of square BCDE.

Proof of the Pythagorean Theorem Triangle Δ CBF has base BF and height BM. Triangle Δ CBF is equal to half the area of rectangle BMNF.

Proof of the Pythagorean Theorem BE is equal to BC because both are sides of the same square. ΔABE is equal ΔFBC by SAS. BA is equal to BF because both are sides of the same square. Angle <EBA is equal to <CBF. Both angles are formed by adding the angle <ABC to a right angle. Thus square BCDE is equal to rectangle BMNF.

World Wide Web java applet for Euclid’s proof. hafv.html

Additional Proofs of the Pythagorean Theorem. Proof by former president James Garfield. eg/Pythagorean.html More than 70 more proofs.

A simple hands on proof for students. Step 1: Cut four identical right triangles from a piece of paper. a b c

A simple hands on proof for students. Step 2: Arrange the triangles with the hypotenuse of each forming a square. a b c b b a a b c c c a Area of large square = (a + b) 2 Area of each part 4 Triangles = 4 x (ab/2) 1 Red Square = c 2 (a + b) 2 = 2ab +c 2 a 2 + 2ab + b 2 = 2ab +c 2 a 2 + b 2 = c 2

Alternate arrangement Area of large square = c 2 Area of each part 4 Triangles = 4 x (ab/2) 1 Purple Square = (a – b) 2 (a – b) 2 + 2ab +c 2 a 2 – 2ab + b 2 + 2ab = c 2 a 2 + b 2 = c 2 b c a – b c c c a

The converse of the Pythagorean Theorem can be used to categorize triangles. If a 2 + b 2 = c 2, then triangle ABC is a right triangle. If a 2 + b 2 < c 2, then triangle ABC is an obtuse triangle. If a 2 + b 2 > c 2, then triangle ABC is an acute triangle.

Cartesian equation of a circle. x 2 + y 2 = r 2 is the equation of a circle with the center at origin.

Pythagorean Fractal Tree Students can create a fractal using similar right triangles and squares. Using right triangles to calculate and construct square roots.

Was Pythagoras a square? The sum of the area’s of the two semi circles on each leg equal to the area of the semi circle on the hypotenuse. The sum of the areas of the equilateral triangles on the legs are equal to the area of the equilateral triangle on the hypotenuse.

Extensions and Ideas for lessons  Does the theorem work for all similar polygons? Is there a trapezoidal version of the Pythagorean Theorem?  Using puzzles to prove the Pythagorean Theorem.  Make Pythagorean trees.  Cut out triangles and glue to poster board to demonstrate a proof of Pythagorean Theorem.  Create a list of Pythagorean triples and apply proofs to specific triples.  Use Pythagorean Theorem with the special right triangles.  Categorize triangles with converse theorem.

References Boyer, Carl B. and Merzbach, Uta C. A History of Mathematics 2 nd ed. New York: John Wiley & Sons, Burger, Edward B. and Starbird, Michael. Coincidences, Chaos, and All That Math Jazz. New York: W.W. Norton & Company, Gullberg, Jan. Mathematics: From the Birth of Numbers. New York: W.W. Norton & Company, Livio, Mario. The Golden Ratio. New York: Broadway Books,

Links berty.Greg/Pythagorean.html