Pythagorean Triple 2: Generating Pythagorean Triples

In the previous math article, we have shown that there are infinitely many Pythagorean triples.  In this article, we are going to discuss a very short but effective strategy in generating Pythagorean Triples.

A Pythagorean triple is the integer triple (a,b, c) satisfying the Pythagorean equation a^2 + b^2 = c^2.

Observe the Pythagorean triples (3, 4, 5) and  (5, 12, 13).  We can see that the hypotenuse is greater than the longer side by 1. From the pattern, we can form the Pythagorean triples (a, b, b + 1) satisfying the equation a^2 + b^2 = (b + 1)^2.

Right triangle with side length 3, 4 and 5 units.

Solving the equation we have a^2 = 2b + 1, which implies that  a = \sqrt{2b +1}.  Now,   2b + 1 is always odd (can you see why?). It follows that in order for a to be an integer, 2b + 1 must be a perfect square. This means, that we are sure that a is an integer, if 2b + 1 is an odd perfect square.

From here, we can generate infinitely many examples of Pythagorean triples.  For example, 49 is an odd perfect square. So plugging it in the equation we have,  2b + 1 = 49, then, we have the triple (7, 24, 25), another Pythagorean Triple.  If we let 2b + 1 = 121, then we have the triple (11, 60, 61).

Now, we found another way to generate Pythagorean Triples.

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GeoGebra Essentials 7 – Using the Keyboard Commands

In the previous tutorials in the GeoGebra Essentials Series, we have discussed how to construct mathematical objects such as points, lines, line segments, circles and other mathematical objects using the mouse and the tools in the Toolbar.  In this post, we  learn how to construct these objects using keyboard commands. In particular, we are going to create an equilateral triangle.

Step by Step Instructions

In this post, we create equilateral triangle ABC. The construction is very similar to GeoGebra Tutorial 2 – Constructing an Equilateral Triangle.

1.) Open GeoGebra. Select Algebra & Graphics in the Perspective panel to open the Algebra & Graphics window

2.) To plot point A with coordinates (1,1), type A = (1,1) in the Input Bar and then press the ENTER key on your keyboard. Now plot B = (5,1).

3.) To construct segment AB, type segment[A,B].

4.) To construct a circle with center A and passing through B, type circle[A,B]. Now, construct a circle with center B passing through A.

5.) Now, to intersect the two circles, type intersect[c,d]. The names c and d are the name given by GeoGebra to the two circles (see Algebra view). Names, usually small or capital letters, are given to each object in GeoGebra. Do not confuse the names with equations.

6.) Next, to polygon ABC, type polygon[A, B, C].

The Final Output

7.) Now, hide the circles and point D by right clicking them and clicking Show Object and we are done.  As we can see, we cannot do everything using keyboard commands. 

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Blog Carnival 7 Deadline of Submission

The Mathematics and Multimedia Blog Carnival is on its seventh edition and will be hosted by Keeping Math Simple. It will be posted on Monday, January 31, 2010. The deadline of submission is on FridayJanuary 28, 2010. You may submit your articles here.

To increase the chance of your article of being published, read the Mathematics and Multimedia Carnival’s Criteria for Selection of Articles. If you missed the first three carnival posts and the latest carnivals, click the links below:

You may also want to read the following posts:

  1. If you don’t about blog carnivals, you may want to read What is a Blog Carnival?
  2. If you are intrested to host the Mathematics and Multimedia Carnival, and have no idea how to go about it, read How to Host a Blog Carnival. Hosting a Blog carnival will definitely popularize your blog.
  3. If you are interested about great articles, read the  Complete List of Let’s Play Math Carnival Posts.

If you are interested to host the math carnival, you can email me here

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