Month: August 2011

The Golden Ratio

High School Geometry, High School Mathematics

Problem: Find point P on segment AB such that AP:PB = AB:AP.

Solution:

If we let r be the length of AP and s be the length of PB, then the ratio above can be written as

\displaystyle\frac{r}{s} =\frac{r+s}{r}

Simplifying the equation we have r^2 = rs + s^2 which is equivalent to r^2-rs - s^2, a quadratic equation in r. So, we solve for r using the quadratic formula with a = 1, b = s and c = -s^2:

r = \displaystyle\frac{-b \pm \sqrt{b^2-4ac}}{2a} = \frac{-s \pm \sqrt{s^2-4(1)(-s^2)}}{2(10)}.

Simplifying this gives us r = \frac{s + s\sqrt{5}}{2}. Dividing both sides by s, we have

\displaystyle\frac{r}{s} = \frac{1 + \sqrt{5}}{2} \approx 1.61803398874989. » Read more

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Geometer’s Sketchpad Essentials 4 – The SSS Triangle Congruence

Geometer's Sketchpad, Software Tutorials

This is the fourth part of the Geometer’s Sketchpad Essentials Series. In this tutorial, we are going to construct another triangle which is congruent to a given triangle using the concept of the SSS triangle congruence.  Recall that the SSS congruence theorem tells us that two triangles are congruent, if their corresponding sides are congruent. In doing the construction, we are going to learn how to use the Ray tool, the Circle tool, and other commands.

1.) Construct triangle ABC.

2.) Next, we construct ray DE.  To do this, click the Straightedge tool box and hold the mouse button to display the other tools. Now, choose the Ray tool.

3.) Click two distinct points on the sketch pad and display the names of the two points. Your sketch should look like the first figure.

4.) Next, we will construct a segment DF which is congruent to AC. To do this, be sure to deselect all the objects by clicking on the vacant part of the sketch pad. Select point D, then select segment AC (do not select the points!), click the Construct menu, and then click Circle By Center+Radius. This will produce a circle with center D and radius equal to the length of AC. » Read more

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Google+: A Facebook replacement?

Web 2.0 Technology

If you have enjoyed Facebook, Google+ is another way to get connected to your family, friends, and colleagues online.  Google+ is a new social networking service lauched by Google. As of this writing, Google+ is on field testing and membership is on an “invite only.”

The things I like most about Google+ is that you can choose “circles” who can view your wall posts. You can share intimate thoughts with friends, and have professional conversations with colleagues without one group reading the message for the other. This is not the case in Facebook yet.

Here are some of the major features of Google+. » Read more

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