Category: High School Geometry

Linking sum of counting numbers to triangle area

High School Algebra, High School Geometry, High School Mathematics

We have discussed how Gauss was able to add the first 100 counting numbers, and we learned ways how to generalize his method.  In this post, we link his method in finding the area of triangles.

Adding the first few counting numbers is easy. However, as the numbers become larger, it becomes harder. According to an anecdote, Gauss at primary school was able to find a clever way of answering the question his teacher asked him:  What is the sum of all the numbers from 1 to 100?

Gauss method was to add the first and the last digit (1 and 100), the second and the second to the last digit (2 and 99), third and the third to the last digit (3 and 98), and found out that the sums were always 101. There are 50 pairs of numbers from 1 to 100 whose sum is 101. So,  the sum of all the numbers from 1 to 100 is (101)(50) = 5050. » Read more

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Properties of Similar Triangles Part 2

High School Geometry, High School Mathematics

This is the third and the conclusion of the Triangle Similarity Series. The two prequels  are 1. Introduction to Similarity and 2. Properties of Similar Triangles (Part 1).

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In the previous post, we have investigated the properties of similar triangles. We have learned that corresponding angles of similar triangles are congruent. In this post, we are going to discuss more about the properties of similar triangles.  If you have not performed the investigation in the previous post, you can use the applet below.

[iframe http://mathandmultimedia.com/wp-content/uploads/2011/09/propertiessimilartriangles.html 539 464]

You would have realized from your exploration of the applet that aside from the angles, there is also something unique about the side lengths of the corresponding sides of the triangles (check the Show/Hide Side Length check box above).  We can verify they have the same ratio.  That is, if triangle ABC is similar to triangle DEF, then the following relationships hold: » Read more

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Undefined terms: What are they?

High School Geometry, Questions and Quandaries

Recall that we have discussed about the meaning of “undefined” in different contexts.  In this post we are going  to discuss the meaning of undefined terms. What do we really mean when we say undefined terms in mathematics?

Polygons (via Wikipedia)

The time we started to learn mathematics, we also started to learn its vocabulary. We have learned about the definitions of terms such as triangles, fractions, polynomials, axioms, etc.  In our years of studies, we also have acquired the skill of being concise and precise in what we say or write.  Like in other fields of endeavors, we need to understand each other and make sure that we mean the same thing when we say a particular word. That is why we need to define them.  For example, when we say p0lygon (see figure above), we mean (loosely) that it is a plane figure bounded  by a finite number of line segments. Notice that from the definition itself, we must also define the other terms used. What do we mean by plane figure or line segment? » Read more

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