Department of Mathematics Education J. Wilson, EMT The Pythagorean Theorem was one of the earliest theorems known to ancient civilizations. This famous theorem is named for the Greek mathematician and philosopher, Pythagoras.
He is credited with many contributions to mathematics although some of them may have actually been the work of his students. The Pythagorean Theorem is Pythagoras' most famous mathematical contribution. According to legend, Pythagoras was so happy when he discovered the theorem that he offered a sacrifice of oxen. The later discovery that the square root of 2 is irrational and therefore, cannot be expressed as a ratio of two integers, greatly troubled Pythagoras and his followers.
They were devout in their belief that any two lengths were integral multiples of some unit length. Many attempts were made to suppress the knowledge that the square root of 2 is irrational. It is even said that the man who divulged the secret was drowned at sea. The Pythagorean Theorem is a statement about triangles containing a right angle. The Pythagorean Theorem states that: "The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides.
Figure 1. According to the Pythagorean Theorem, the sum of the areas of the two red squares, squares A and B, is equal to the area of the blue square, square C. Although Pythagoras is credited with the famous theorem, it is likely that the Babylonians knew the result for certain specific triangles at least a millennium earlier than Pythagoras. It is not known how the Greeks originally demonstrated the proof of the Pythagorean Theorem. Therefore, the square on c is equal to the sum of the squares on a and b.
Burton There are many other proofs of the Pythagorean Theorem. One came from the contemporary Chinese civilization found in the oldest extant Chinese text containing formal mathematical theories, the Arithmetic Classic of the Gnoman and the Circular Paths of Heaven. 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. Find all right triangles whose sides are of integral length, thus finding all solutions in the positive integers of the Pythagorean equation:.
The formula that will generate all Pythagorean triples first appeared in Book X of Euclid's Elements :. 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. Students 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. 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. He proved these equalities using the concept of similarity. Since the sum of the areas of the two rectangles is the area of the square on the hypotenuse, this completes the proof. Euclid was anxious to place this result in his work as soon as possible. However, since his work on similarity was not to be until Books V and VI, it was necessary for him to come up with another way to prove the Pythagorean Theorem.
Thus, he used the result that parallelograms are double the triangles with the same base and between the same parallels. Draw CJ and BE. The two triangles are congruent by SAS. The same result follows in a similar manner for the other rectangle and square. Katz, Click here for a GSP animation to illustrate this proof.
The Pythagoras theorem states that if a triangle is right-angled 90 degrees , then the square of the hypotenuse is equal to the sum of the squares of the other two sides. Hence, any triangle with one angle equal to 90 degrees will be able to produce a Pythagoras triangle. Pythagoras theorem was introduced by the Greek Mathematician Pythagoras of Samos. He was an ancient Ionian Greek philosopher. He started a group of mathematicians who works religiously on numbers and lived like monks.
Finally, the Greek Mathematician stated the theorem hence it is called by his name as the "Pythagoras theorem. Although Pythagoras introduced and popularised the theorem, there is sufficient evidence proving its existence in other civilizations, years before Pythagoras was born.
The oldest known evidence dates back to between 20th to 16th Century B. C in the Old Babylonian Period. The Pythagoras theorem formula states that in a right triangle ABC, the square of the hypotenuse is equal to the sum of the square of the other two legs.
Pythagoras theorem can be proved in many ways. Some of the most common and most widely used methods are by using the algebraic method proof and using the similar triangles method to solve them. Let us have a look at both of these methods individually in order to understand the proof of this theorem. Algebraic method proof of Pythagoras theorem will help us in deriving the proof of the Pythagoras Theorem by using the values of a, b, and c values of the measures of the side lengths corresponding to sides BC, AC, and AB respectively.
Consider four right triangles ABC where b is the base, a is the height and c is the hypotenuse. The area of the square so formed by arranging the four triangles is c 2. Hence Proved. Two triangles are said to be similar if their corresponding angles are of equal measures and their corresponding sides are in the same ratio.
Also, if the angles are of the same measure, then we can say by using the sine law, that the corresponding sides will also be in the same ratio. Hence, corresponding angles in similar triangles will lead us to equal ratios of side lengths.
Right triangles follow the rule of the Pythagoras theorem and they are called Pythagoras theorem triangles. The length of all the three sides are being collectively called Pythagoras triples. For example, 3, 4, and 5 can be called as one of the sets of such triangles.
There are a lot more right-angled triangles which are called Pythagoras triangles. And these squares are known as Pythagoras squares. Though it is necessary to learn the basic concepts such as theorem statements and their mathematical representation, we would be more curious in understanding the applications of the Pythagoras theorem which we face in day-to-day life situations.
Most architects use the technique of the Pythagorean theorem to find the value as well as when length or breadth are known it is very easy to calculate the diameter of a particular sector. It is mainly used in two dimensions in engineering fields. We are more familiar with face recognition nowadays it reduces the turmoil in investigating the crimes in the security areas.
It undergoes the concept of the Pythagorean theorem that is, the distance between the security camera and the place where the person is noted is well projected through the lens using the concept.
As the main concept indicates if the cardboards being square can be made into a triangle easily by cutting diagonally then very easily the Pythagoras concept can be applied. Most woodworks are made on the strategy which makes it easier for designers to proceed.
It's a very amazing fact but people traveling in the sea use this technique to find the shortest distance and route to proceed to their concerned places.
Usually, surveyors use this technique to find the steep mountainous region, knowing the horizontal region it would be easier for them to calculate the rest using the Pythagoras concept. The fixed distance and the varying one can be looked through a telescope by the surveyor which makes the path easier. If you want to build more conceptual knowledge with the help of practical illustrations try Pythagoras Theorem Worksheets. Also, check out few more interesting articles related to Pythagoras Theorem for better understanding.
Example 1: Consider a right-angled triangle. The measure of its hypotenuse is 16 units. One of the sides of the triangle is 8 units. Find the measure of the third side using the Pythagoras theorem formula? Example 2: Julie wanted to wash her building window which is 12 feet off the ground. She has a ladder that is 13 feet long.
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