The proof of the Pythagorean Theorem that was inspired by a figure in this book was included in the book Vijaganita, (Root Calculations), by the Hindu mathematician Bhaskara. Bhaskara's only explanation of his proof was, simply, "Behold".
These proofs and the geometrical discovery surrounding the Pythagorean Theorem led to one of the earliest problems in the theory of numbers known as the Pythgorean problem.
The Pythagorean Problem:
Find all right triangles whose sides are of integral length, thus finding all solutions in the positive integers of the Pythagorean equation:
The three integers (x, y, z) that satisfy this equation is called a Pythagorean triple.
Some Pythagorean Triples:
x y z
3 4 5
5 12 13
7 24 25
9 40 41
11 60 61
The formula that will generate all Pythagorean triples first appeared in Book X of Euclid's Elements:
where n and m are positive integers of opposite parity and m>n.
In his book Arithmetica, Diophantus confirmed that he could get right triangles using this formula although he arrived at it under a different line of reasoning.
The Pythagorean Theorem can be introduced to students during the middle school years. This theorem becomes increasingly important during the high school years. It is not enough to merely state the algebraic formula for the Pythagorean Theorem. Studen辣^文-论~文.网http://www.751com.cn ts need to see the geometric connections as well. The teaching and learning of the Pythagorean Theorem can be enriched and enhanced through the use of dot paper, geoboards, paper folding, and computer technology, as well as many other instructional materials. Through the use of manipulatives and other educational resources, the Pythagorean Theorem can mean much more to students than just
and plugging numbers into the formula.
The following is a variety of proofs of the Pythagorean Theorem including one by Euclid. These proofs, along with manipulatives and technology, can greatly improve students' understanding of the Pythagorean Theorem.
The following is a summation of the proof by Euclid, one of the most famous mathematicians. This proof can be found in Book I of Euclid's Elements.
Proposition: In right-angled triangles the square on the hypotenuse is equal to the sum of the squares on the legs.
Figure 2
Euclid began with the Pythagorean configuration shown above in Figure 2. Then, he constructed a perpendicular line from C to the segment DJ on the square on the hypotenuse. The points H and G are the intersections of this perpendicular with the sides of the square on the hypotenuse. It lies along the altitude to the right triangle ABC. See Figure 3.
Figure 3
Next, Euclid showed that the area of rectangle HBDG is equal to the area of square on BC and that the are of the rectangle HAJG is equal to the area of the square on AC. He proved these equalities using the concept of similarity. Triangles ABC, AHC, and CHB are similar. The area of rectangle HAJG is (HA)(AJ) and since AJ = AB, the area is also (HA)(AB). The similarity of triangles ABC and AHC means