Pythagorean Theorem - Real World Application

Posted by David Wetzel

Pythagorean Theorem

The Pythagorean Theorem is used any time you have a right triangle in which you know the length of two sides and want to find the third side.

The formula is: a² + b² = c²

There are many strategies for getting students to learn how to solve a problem related to this theorem. Many of these problems are boring, procedural driven, and lacking in connection to real world applications.

In the view of many this is often necessary because they do not have the time to teach it any other way. Curriculum, testing, and No Child Left Behind pressures cause these views.

However, there are other strategies to teach the same concept and yet make it more interesting for students. The critical aspect of this strategy is to make connections to their world.

Students always ask:

• When will I every use the Pythagorean Theorem?
• Why do need to learn this? It is boring!

The answer is, in any kind of job that deals with triangles. For example, carpenters, engineers, architects, construction workers, those who measure and mark land, artists, and designers of many sorts need to know it.

Problem Solving Application

The following are problem is more interesting and designed to find the length of a² and b², when only c² is known.

Shopping for a New TV

Jacob and Denise, brother and sister, are trying to get their parents to buy them a new TV. Their parents have agreed; however, they are limited to a size 32 inch TV.

Now they must go shopping to find the one they want. At their local electronics store, they discover that 42 inch TVs come in a lot shapes and sizes.

This does not seem to make since, until the salesman tells that 42 inches is diagonal measurement that is the normal way of sizing TVs.

They decide to compare the difference between a 32 inch standard TV and 32 inch flat screen HDTV. Jacob and Denise both agreed that they should buy the TV with greatest viewing area.

After looking a both TV types, they predicted that the flat screen HD TV had the greatest viewing area.

How would you determine which TV has the greatest viewing area?

Answer

TV Problem: the key to this problem is the aspect ratio of the TVs. The Standard TV has an aspect ratio of 4:3, while the Flat Screen HDTV has an aspect ratio of 21:9.

First they must find out what each unit equals (for example in the Standard TV is 4 units wide and 3 units high - aspect ration 4:3) in the aspect ratio for each TV by using the Pythagorean Theorem.

Next they must find the area of each TV using A = w x h.

Standard TV Area = 491.56 in²

Flat Screen HD TV Area = 370.44 in²

Jacob and Denise now must decide which TV they really want, because their prediction was wrong.

——

This problem not only makes connections to something that students are actually interested in, they also make connections to other math concepts. Pythagorean Theorem now has a real world application to students and they will see its value.

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Geometry of Triangles: Interactive Problem Solving

Posted by David Wetzel

Geometry of Triangles

These resources offer online opportunities to explore the geometry of triangles.

From ideas of congruence to reflections to relationships among the triangle’s angles and sides.

Interactive Geometry Websites

Congruence - Using virtual manipulatives, students can arrange sides and angles to construct congruent triangles using Congruence of Triangles.

Angles - This interactive problem solving exploration of triangles begins at the beginning—with angles and their classification. Students can practice their understanding and then move on to construct triangles and consider the sum of the angles of any triangle. Finally, they explore the special relationship among the sides of a right triangle—the Pythagorean theorem, demonstrated here through a Java slide show using Geometry of Angles.

Equilateral - Students learn the theorems of triangle congruence using a challenging problem solving application. Here students are presented with an intricate figure showing two overlapping equilateral triangles. Because this resource is an applet, students can rotate the figure and easily see that two triangles in the figure are congruent. The challenge is to prove the triangles are congruent using Two Equilateral Triangles.

Transformation and Reflection - Students can manipulate one of six geometric figures on one side of a line of symmetry and observe the effect on its image on the other side. A triangle may be selected and then translated and rotated. The line of symmetry can be moved as well, even rotated, giving more hands-on experience with reflection as students observe the effect on the image of the triangle using Transformations and Reflection.