## Task 4 – Point of Similarity

Shown below are $\triangle STU$ and $\triangle MNO$.

Questions

1.) Are the two triangles similar? Why?

2.) What is the ratio of the lengths of their corresponding sides?

3.) Use the Line tool to connect the corresponding vertices of the two triangles. Did your observations in Task 2 and Task 3 hold?

4.) Make a conjecture about the lines connecting the corresponding vertices of similar triangles.

[toggle title=’Explanation’ title_font_size=’14px’]If two triangles are similar and their corresponding sides are parallel, then the line connecting their corresponding vertices will intersect at a point. This point is called the center of similarity or point of similarity.

In Task 3, $\triangle ABC \sim \triangle A'B'C'$ and their corresponding sides are parallel. The three lines connecting the corresponding vertices intersect point $P$ as shown above. In Task 4, $\triangle STU \sim \triangle MNO$, but their corresponding sides are not parallel, so the three lines connecting their corresponding vertices do not intersect at a point.

[/toggle]

## Task 3 – Connecting Vertices of Similar Triangles

Shown in the applet below are $\triangle EFG$ and $RST$.

Questions

1.) Are the two triangles similar? Why?

2.) Use the Line tool to connect the corresponding vertices of the two triangles. What do you observe about the lines?

3.) Draw another triangle that is similar to $\triangle EFG$  and name it $\triangle HJI$. Connect their corresponding vertices. Did your observation in 2 also hold?

## Task 2 – Drawing Similar Triangles (Part 2)

In the applet below, $A'B' = 2AB$, $A'C'= 2AC$, and $B'C'= 2BC$. Use the Segment tool to draw $\triangle PQR$ such that $PQ = 3AB$, $PR= 3AC$, and $QR = 3BC$.

Questions

1.) How did you draw ∆PQR?

2.) What do you observe about the corresponding angles of the ∆ABC, ∆A’B’C’ and ∆PQR?

Discussion

In Task 1 and Task 2, you have observed that it is easier to draw similar triangles if you make the corresponding sides of the triangles parallel. In addition, you have also observed that the corresponding angles of the triangles are congruent. For example,

$\angle A \cong \angle A'$

$\angle B \cong \angle B'$

and

$\angle C \cong \angle C'$

In drawing the triangles, you made sure that the ratio of the lengths of corresponding sides is equal. In other words, you made the length of the corresponding sides proportional. For instance, the ratio of the lengths of the corresponding sides of to those of is 2. That is,

$\frac{A'B'}{AB'} = 2$

$\frac{A'C'}{AC'} = 2$

$\frac{B'C'}{BC} = 2$

Two triangles whose corresponding angles are congruent are called similar triangles. The lengths of the corresponding sides of similar triangles are proportional.

***

## Task 1 – Drawing Similar Triangles (Part 1)

In the applet below, given $\triangle ABC$, use the Segment tool to draw $\triangle A'B'C'$  such that $A'B' = 2AB$, $A'B' = 2AC$, and $B'C' = 2BC$.

Questions

1.) How did you draw $\triangle A'B'C'$?

2.) What do you observe about the angles of the two triangles?

3.) Draw other triangles satisfying the same conditions and see if your observations still hold.

In your lesson on Triangle Congruence, you have learned about corresponding angles and corresponding sides. In the two triangles above, the pairs of corresponding angles are $A$ and $A'$, $B$ and $B'$, and $C$ and $C'$. The pairs of corresponding sides are $AB$ and $A'B'$, $BC$ and $B'C'$ and $AC$ and $A'C'$.

## Visual Proof of Pythagorean Theorem

Shown below is a triangle whose sides form squares with sides a, b, and c.

1.) Move slider p to the extreme right.

What is the relationship among the squares when p was on the extreme left and the quadrilaterals when p was on the extreme right?

2.) Move slider q to the extreme right.

What is the relationship among the quadrilaterals when q was on the extreme left and when q was on the extreme right?

3.) Move slider r to the extreme right.

What is the relationship among the quadrilaterals when r was on the extreme left and when r was on the extreme right?

4.) Based on your exploration, what is the relationship among the areas of the squares with side lengths a, b, and c?

5.) Using the variables, a, b, and c write an equation about the relationship you found in 4.

## Dice Rolling Simulation

Click the Roll Dice button and record the sum of the number of dots. Do this for 20 times and tally the result.

Questions1.) What do you observe?
2.) Which sum occurs more often? less often?
3.) Do the simulation for 30 times, 40 times, 50 times, 100 times. Are your observations still the same?
4.) Explain why your observations are such.

## The Infinite Chocolate Puzzle

View the video below and observe what happens. Can you tell where did the extra chocolate come from?

Now, explore the GeoGebra applet below by moving point A (up and down) and moving points B and C along the diagonal.

1. What do you observe about the video and the applet?
2. Are the video and applet similar?
3. Where did the extra chocolate come from?

Thumbnail

Created with GeoGebra by Guillermo Bautista

## The Date Problem

You and a friend arrange to meet between 12:00 and 1:00 in the afternoon. After a week neither of you remembers the exact meeting time. As a result, it is possible for you arrive at random between 12:00 and 1:00 and waits exactly 15 minutes for your friend to arrive. After 15 minutes, each of you leaves if the other person has not arrived. What is the probability that the two of you will meet?

Click the Play button at the bottom left of the applet.

Explore!

1.) Click Pause and see the exact time the two friends arrive.
2.) Refresh the browser to delete the point traces.
3.) Click Pause and drag the sliders to customize the time arrival.

***

Problem by Erlina R. Ronda
Created with GeoGebra by Guillermo P. Bautista Jr

Thumbnail

## GeoGebra Tutorial: Exporting GeoGebra Worksheet to HTML 5

GeoGebra worksheets cannot run on devices that do not support Java. Examples of such devices are Apple’s iPads. To address this limitation, GeoGebra has created an export feature to HTML5. In this tutorial, we learn how to export a GeoGebra worksheet to HTML5.
Steps in Exporting a GeoGebra Worsheet to HTML5

1. Open the GeoGebra worksheet that you want to export to HTML5.
2. Click the Filemenu, select Export, and then select Dynamic Worksheet as Webpage (html).

2. In the Dynamic Worksheet dialog box, select Export as Webpage button.
3. Type the Title of the web page and the texts needed above and below the construction.
4. Select the Advancedtab and be sure to check the Export to HTML only and Remove Line Breaks check boxes
5. Select File: html in the File section
6. Click the Exportbutton

The exported HTML file can be used as a web page or embedded to websites or blogs. The HTML5 page does not need Java in order to run.

## The Chebyshev Walking Machine

The Chebyshev linkage is a mechanical linkage that converts rotational motion to approximate straight-line motion. It was invented by the 19th century mathematician Pafnuty Chebyshev who studied theoretical problems in kinematic mechanisms.

One of these linkages is shown above. It is called the Chebyshev walking machine. Notice that the mechanism seems to reproduce the legs of an animal. Click the Play button to start the animation.
1 2 3 4 8