kemunto ondande, alex nelson, and hunter brehm
TRANSCRIPT
Proofs of the Pythagorean Theorem
Kemunto Ondande, Alex Nelson, and Hunter Brehm
Part A Report on at least three other proofs of the
Pythagorean theorem besides the ones illustrated in
this chapter. If you find further interesting historical
information about this great theorem, report on that
too.
Garfield
● James A. Garfield was the twentieth president of the United States.
● Before becoming president, he was a professor at Western Reserve Eclectic
Institute, now called Hiram College, in Hiram, Ohio.
● In addition to being a mathematician, he had unusual abilities such as being
able to write Greek in the left hand while writing Latin in the right hand.
● Although he was aware of many proofs of the Pythagorean Theorem, he was
able to construct his own unique proof.
● He developed his proof in 1876 while working as a Member of Congress.
Garfield’s Proof
His proof consists of a diagram of a
trapezoid with a parallel sides a and b and
height a + b. Garfield looked at the area of
the diagram in two different ways: as that of
a trapezoid and as that of three triangles,
two of which are congruent.
Garfield’s Proof
Proof:
● Start with a right triangle with legs a and b and hypotenuse c.
● Extend leg a by b units and construct a duplicate right triangle along this
extension.
● Upper leg a is parallel to the original leg b since, in a plane, if a line is
perpendicular to each of two lines, then the two lines are parallel.
● Draw a segment XY to close the figure.
● The resulting quadrilateral is a trapezoid with bases a and b and altitude a + b.
● The trapezoid is composed of two congruent right triangles and right triangle
XYZ. Triangle XYZ is isosceles since two of its sides have length c. Angle
XZY is a right angle since ∡ 1 + ∡ XZY + ∡ 2 = 180˚ and ∡ 1 + ∡ 2 = 90˚.
● Therefore, the area of the trapezoid equals the sum of three right triangles of
which it is composed, two of which are congruent.
Garfield’s Proof
Algebraic expression:
● Area of a trapezoid = 1
2(sum of the parallel sides)(the height of the
trapezoid)
● 𝐴 =1
2(𝑎 + 𝑏)(𝑎 + 𝑏)
● Area of the scalene right triangle: 𝐴𝑠 =1
2𝑎𝑏
● Area of the isosceles right triangle: 𝐴𝑖 =1
2𝑐2
● 𝐴𝑡 = 2 𝐴𝑠 + 𝐴𝑖
● 𝐴 =1
2𝑎 + 𝑏 𝑎 + 𝑏 = 2
1
2𝑎𝑏 +
1
2𝑐2
● 𝑎 + 𝑏 𝑎 + 𝑏 = 2𝑎𝑏 + 𝑐2
● 𝑎2 + 2𝑎𝑏 + 𝑏2 = 2𝑎𝑏 + 𝑐2
● 𝑎2 + 𝑏2 = 𝑐2
Bhaskara
Bhaskara was an Indian mathematician in the 12th century that proved the Pythagorean Theorem.
He developed two proofs of the Pythagorean Theorem:
1st Proof: consists of placing congruent right triangles around a small square in such a way that the
hypotenuses of each triangle form a larger square on the outside
2nd Proof: consists of drawing an altitude of a right triangle to hypotenuse c and then using similarity
between the triangle and the two smaller triangles within it, formed by the altitude, to prove the
theorem
Bhaskara’s 1st Proof of Pythagorean Theorem
● 𝐴 = 𝑐2
● Area of the blue triangles = 41
2𝑎𝑏 = 2𝑎𝑏
● Area of the yellow square = (𝑏 − 𝑎)2
● Area of the big square = 41
2𝑎𝑏 + (𝑏 − 𝑎)2
= 2ab + 𝑏2 − 2𝑎𝑏 + 𝑎2
= 𝑎2 + 𝑏2
𝑐2 = 𝑎2 + 𝑏2
Bhaskara’s 2nd Proof of Pythagorean Theorem
Proof:
● Draw an altitude on the hypotenuse of a right triangle
● Show that the right triangle & one of the two congruent triangles that the
altitude formed were similar.
● We will let r be 𝑐1 for triangle ACE and let s be 𝑐2 for triangle CBE.
● Prove the triangle ABC ~ triangle CBE
● Show triangle ABC ~ triangle ACE
● Add the two similar triangles: 𝑠𝑐 + 𝑟𝑐 = 𝑎2 + 𝑏2
● Since we know that 𝑐 = 𝑠 + 𝑟. We can solve this algebraically.
● So, 𝑠𝑐 + 𝑟𝑐 = 𝑎2 + 𝑏2
𝑐 𝑠 + 𝑟 = 𝑎2 + 𝑏2
𝑐2 = 𝑎2 + 𝑏2
Part BA Pythagorean triple (a, b, c) of positive integers
satisfying the Pythagorean equation. The triple is
primitive if the integers have no common factor. A
general Pythagorean triple is a positive integer
multiple of a primitive one (cancel the gcd). Find the
polynomials p, q, r of degree 2 in two integer
variables such that every primitive Pythagorean
triple is given by a = p(m, n), b = q(m, n), and c =
r(m, n) and conversely these equations provide a
primitive Pythagorean triple for every pair of
unequal relatively prime positive integers (m, n).
Search the web for further results on Pythagorean
triples and report on the results you find most
interesting.
Polynomials for Primitive Pythagorean Triple
● The formula for the Pythagorean Theorem can be easily converted into the equation for the unit circle.
Pythagorean Theorem:
Further Results on Pythagorean Triples
● Many years ago before the Pythagoras, the Babylonians knew how to create Pythagorean triples using natural
numbers: (a,b,c) such that 𝑎2 + 𝑏2 = 𝑐2.
● If (m,n) is a pair of natural numbers with m > n, then: (a,b,c) = (𝑚2 − 𝑛2, 2𝑚𝑛,𝑚2 + 𝑛2) is a Pythagorean triple.
[(m,n) is consider the generator of (a,b,c)]
● If (a,b,c) is any Pythagorean triple and d is any natural number, then (da,db,dc) is consider a Pythagorean triple.
● Another way in generating primitive Pythagorean triples is by the following: 𝑎, 𝑏, 𝑐 = (𝑟𝑠,𝑟2−𝑠2
2,𝑟2+𝑠2
2) where (r, s)
is a pair of odd integers , greater common denominator (gcd) of (r, s)=1, and r > s ≥ 1
● A solution to the Diophantine equation 𝑥2 + 𝑦2 = 𝑧2 (Pythagorean triples)
● There are no Pythagorean triples in which both legs of a triangle are odd.
○ Theorem: If (a,b,c) is a Pythagorean triple, then at least one of the positive integers a or b is even.
References
Ballard, William R. Geometry. Philadelphia, Saunders, 1970.
Bogomolny, Alexander. Pythagorean Theorem and Its Many Proofs, Alexander Bogomolny,
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Eckert, E. (1992). Primitive Pythagorean Triples. The College Mathematics Journal, 23(5), 413-417. doi:10.2307/2686417
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jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/HeadAngela/essay1/Pythagorean.html.
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Garfield’s Proof of the Pythagorean Theorem | Mathematical Association of America, Mathematical Association of America, Feb.
2016, https://www.maa.org/press/periodicals/convergence/mathematical-treasure-james-a-garfields-proof-of-the-pythagorean-theorem
Marshall, D. C., Starbird, M. P., & Odell, E. (2007). Number Theory Through Inquiry. Washington, DC: Mathematical Association of
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