StudentShare solutions

The Pythagorean Theorem - Essay Example

Extract of sample The Pythagorean Theorem

The area of such a rectangle is a times b: ab. Therefore the four triangles together are equal to two such rectangles. Their area is 2ab.

As for the square whose side is c, its area is simply c. Therefore, the area of the entire square is
c + 2ab . . . . . .(1)
At the same time, an equal square with side a + b (Fig. 2) is made up of a square whose side is a, a square whose side is b, and two rectangles whose sides are a, b. Therefore the area of that square is
a + b + 2ab
But this is equal to the square formed by the triangles, line(1):
a + b + 2ab = c + 2ab.
Therefore, on subtracting the two rectangles -- 2ab -- from each square, we are left with
a + b = c.
This is the Pythagorean Theorem

Proof using similar triangles

The Pythagorean theorem, is based on the proportionality of the sides of two similar triangles.
Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. We draw the altitude from point C, and call H its intersection with the side AB. The new triangle ACH is similar to our triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well. By a similar reasoning, the triangle CBH is also similar to ABC. The similarities lead to the two ratios..: As

so

These can be written as

Summing these two equalities, we obtain

In other words, the Pythagorean theorem:

The Arabian mathematician Thabit ibn Kurrah

A clever proof by dissection which reassembles two small squares into one larger one was given by the Arabian mathematician Thabit ibn Kurrah (Ogilvy 1994, Frederickson 1997).

Proof by Perigal

Another proof by dissection is due to Perigal (left...
Therefore the four triangles together are equal to two such rectangles. Their area is 2ab.
At the same time, an equal square with side a + b (Fig. 2) is made up of a square whose side is a, a square whose side is b, and two rectangles whose sides are a, b. Therefore the area of that square is
Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. We draw the altitude from point C, and call H its intersection with the side AB. The new triangle ACH is similar to our triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well. By a similar reasoning, the triangle CBH is also similar to ABC. The similarities lead to the two ratios..: As
Another proof by dissection is due to Perigal (left figure; Pergial 1873; Dudeney 1970; Madachy 1979; Steinhaus 1999, pp. 4-5; Ball and Coxeter 1987). A related proof is accomplished using the above figure at right, in which the area of the large square is four times the area of one of the triangles plus the area of the interior square. From the figure d=b-a, so
Perhaps the most famous proof of all times is Euclid's geometric proof , although it is neither the simplest nor the most obvious. ...Show more

Summary

In mathematics, the Pythagorean Theorem is a relation in Euclidean geometry among the three sides of a right triangle. The theorem is named after the Greek mathematician Pythagoras who lived in the 6th century B.C.
"In any right angle triangle, the area of the square of the side opposite the right angle i.e…
Author : jaidakoss
Save Your Time for More Important Things
Let us write or edit the essay on your topic
"The Pythagorean Theorem"
with a personal 20% discount.
Grab the best paper

Check these samples - they also fit your topic

Option 1: Dictionary of Terms: Epistemology and Theology
Empiricism is the rational outcome of positivism in that it rejects deductive reasoning (Mende, 189). In other words, the doctrine of empiricism demands a restraint from the employment of non-scientific conjectures (Doyal and Harris, 2). By and large, empiricism refers to a philosophical principle that believes that knowledge comes no more than or mainly from physical experience.
4 pages (1000 words) Essay
A Person Who Has Made Some Contribution to Mathematics
Mathematics is an extremely varied subject and there are numerous individuals all through the course of history who have made a contribution to the field of mathematics in one way or another. The list of those who have contributed to the field of mathematics is extremely long.
4 pages (1000 words) Essay
Pythagorean triplets
2 pages (500 words) Essay
Education Testing Assessment Selection and Development
In a multiple-paragraph composition, explain the advantages and disadvantages of America's two-party system. Include an introduction, supporting body paragraphs with good details and examples, and a conclusion. Use a thesis statement in your introduction and topic sentences in your supporting paragraphs.
2 pages (500 words) Essay
Math Standards
Once students are able to commit these to memory, their ability to solve longer problems with ease becomes apparent, and students can easily undertake multiple step problems using all of these skills. As a student progress through their elementary, middle school, and high school educational lives they should acquire and reach a certain number of these skill benchmarks.
2 pages (500 words) Essay
Education - Testing: Assessment Selection and Development #1
ement required (gallons, inches, etc.), the number of decimal places required, or any other direction necessary for the candidate to know exactly how to formulate an answer. In a multiple-paragraph composition, explain the advantages and disadvantages of America’s two-party
4 pages (1000 words) Essay
Brief Summary of Historical Orientation of Mathematics
As far as the history of mathematics is concerned, the same can be traced back to 3000 BCE. The Base-60 place value system started to develop within Mesopotamia and the Egyptians used the base-10 for grouping different numbering systems. This was the time when Harappan civilization was getting its roots firm within India.
1 pages (250 words) Essay
Pythagorean triples
Hence, given any two sides of a right triangle (triangle with a 90-degree angle) the unknown side can always be solved by algebraically manipulating the equation presented above. A special case of such triangles is when
2 pages (500 words) Essay
Pythagorean Thagorean equations
In all the triples identified, it was verified that a, b, and c are in the exact proportion and relationship in accordance with the Pythagorean Theorem equation. The two numbers of the
1 pages (250 words) Essay
Pythagorean Theorem
Unlike any other mathematical theorem, the Pythagorean Theorem is supported by both geometric and algebraic proofs. This paper will discuss the Pythagorean Theorem in detail. The theorem is a statement about the three sides of a
3 pages (750 words) Essay
Click to create a comment
Let us find you another Essay on topic The Pythagorean Theorem for FREE!