Category: Geometry

In my previous post, I created sets of regular polygons in GeoGebra by setting a parameter of the polygons equal to a constant. In this post, I will show the mathematics for determining the side length given a particular parameter. The values I calculated were side length, radius length, apothem length, area, height, and width….

I recently found myself creating a set of regular polygons for a worksheet. I used GeoGebra to create them, and then free-handed the zoom in order to get them consistently sized. This led me to wonder what “consistently sized” would mean when it comes to polygons. There are six basic values of a regular n-gon:…

In my last post, I pointed out that SSA is in fact sufficient for determining all three sides and angles under certain conditions. In this post, I will specify those conditions, with illustrations. Given two noncollinear segments \(\overline{S_1}\) and \(\overline{S_2}\) and angle \(\angle A\), where  \(\overline{S_1}\)’s two endpoints are the vertex of \(\angle A\) and an…

There is a standard litany of theorems involving proving triangle congruence that has remained largely unchanged since my high school days. I was told that, to prove that two triangles are congruent, we need three pieces of information. The abbreviations were given as SSS, SAS, AAS, and ASA. Astute students would ask about SSA (or…

I read an article today on the six basic trigonometric functions, and I thought there was a particularly important insight that I wanted to present in my own words. When I was in school, we learned the six basic trigonometric functions. Since I’ve been teaching, I’ve noticed that only three of these are emphasized: Sine,…

Introduction In this entry, I’m going to demonstrate the use of GeoGebra to estimate a value for a fairly tricky trigonometry problem, then illustrate how to find the value using trigonometry and an appeal to WolframAlpha. In so doing, I hope to also illustrate the eight basic standards for mathematical practice within Common Core. Here…

Congruence As I discussed in an earlier post, there are two basic definitions of geometric congruence that are presented to students. The first is based on measurement: Definition 1. Two objects are congruent if all of their measurements are the same, and in the same order. That is, two segments are congruent if they’re the…

A standard high school geometry textbook talks about congruence in terms of three types of objects: Line segments, angles, and polygons. Congruence is then defined in terms of measurable parameters: “Two figures are congruent if they have the same size and the same shape” (Carnegie’s Bridge to Algebra Student Text, 2008, p. G-9). Math Open Reference…

Introduction In my previous post, I included this image, which I’d made in GeoGebra. The image satisfies the conditions of the problem: \(AD\) is tangent to \(\odot P\) and \(\overline{BC} \cong \overline{AD}\). In order to create this image, I created a dynamic GeoGebra image where A, B, P and the radius of P can be…

The Golden Ratio By definition, the Golden Ratio is a ratio involving overlapping line segments. Given collinear points A, B, and C, such that B is between A and C, if the ratio between the two subsegments is the same as the ratio between the entire segment and the longer segment, then that ratio is…