GeoGebra Tutorial 27 – Animation and Epicycle

This is the 27th tutorial of the GeoGebra Intermediate Tutorial Series. If this is your first time to use GeoGebra, you might want to read the GeoGebra Essentials Series.

In this tutorial, we rotate a circle about the center of another circle tangent to it using the animation feature of GeoGebra. Along the rotating circle, we will also rotate a point on its circumference about its center (see red point in the diagram). The path of this point is called the epicycloid.

If you want to follow this tutorial step-by-step, you can open the GeoGebra window in your browser by clicking here.  You can view the output of this tutorial here. » Read more

The Inscribed Angle Theorem and Its Applications

An inscribed angle is an angle formed by two chords of a circle with the vertex on its circumference.  In the first circle in Figure 1, segments AB and AC are chords of a circle and the vertex A is on its circumference. Hence, angle A is an inscribed angle.  In the second circle in Figure 1, angle Q is also an inscribed angle.

Angles formed by the two radii of a circle are called central angles. The vertex of a central angle is on the center of the circle. In the Figure 1 below, angles BOC and SOT are central angles.

The arc determined by the endpoints of two chords or two radii, and is opposite of the angle, is called the intercepted arc. In the first circle in Figure 1, arc BC, denoted by the red curve, is the intercepted arc of angle BAC (or angle BOC). Similarly, in the second circle, arc ST denoted by the green curve, is the intercepted arc of the central angle SOT.  Saying the other way around,  angle BAC is the subtended angle of arc BC and and angle SOT is the subtended angle of arc ST.

The Inscribed Angle Theorem

In this article, we are going to discuss the relationship between an inscribed angle and a central angle (I have created a GeoGebra applet about it) having the same intercepted arc.  This is shown in the first circle in Figure 1. Angle BAC and angle BOC have the same intercepted arc BC. Therefore, if a central and an inscribed angle have the same intercepted arc, the measure of that central angle is twice that of the measure of the inscribed angle.

 

Figure 1 – Inscribed angles, central angles and their intercepted arcs.

 

One strategy to solve a problem in geometry is to sometimes draw lines from one point to another. These lines drawn are called auxiliary lines. » Read more

A Gentle Introduction to Probability

Sisters’ Dilemma: Who’s going to grandma?

Issa and Ishi go to their grandmother every weekend. Next weekend, however, their mother will be leaving for an important errand, so one of them has to stay home. Both sisters wanted to go, so to decide, they agreed on a toss coin. If the result is head, then Issa will go, if tail, Ishi will.

Tossing a coin would only result to two possible outcomes: a head or a tail. Assuming that the coin is fair, the chance of Issa winning the toss coin is 1 out of 2, or 50 percent. From the context, intuition tells us that the chance of Issa winning is the ratio of the number of outcomes favorable to her (that is getting a head) to the total number of possible outcomes (head and tail). Obviously, Ishi has the same chance of winning. » Read more

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