In Geometer’s Sketchpad Tutorial 1, we have learned how to construct an equilateral triangle. In this tutorial, we will explore the relationship of a triangle with a moving vertex on a line parallel to its base. We will construct the drawing shown in* Figure 1*. In *Figure 1*, ** PQR **is a triangle,

**is parallel to**

*QC***and**

*PR***is the altitude of triangle**

*QD**.*

**PQR**What happens if we move ** Q** along

**with points**

*QC***and**

*P***fixed?**

*R*In constructing the figure, we will learn the following skills:

- construct perpendicular and parallel lines
- construct interior of a triangle
- measure the distance between two points
- measure the area of a polygon
- measure the length of a segment
- rename objects

Follow the steps below to construct the Dancing Triangle.

**Part I: Constructing the Dancing Triangle
**

1.) Open Geometer’s Sketchpad.

2.) Click the arrow at the *straightedge tools* and choose *line tool*.

3.) Click two distinct locations on the drawing area to construct line ** AB. **Notice that Sketchpad names the objects in alphabetical order

*.*

4.)Display the label of the two points by selecting both of them, clicking the **Display** menu from the menu bar and selecting **Show labels** from the list. If you have not read tutorial 1, we* select* an object by clicking the *arrow tool* and clicking the object. The arrow tool is used in selecting and moving objects.

5.) Next, we change the name of the two points to ** P** and

**. To rename point**

*R***to**

*A***, select point**

*Q***, click the**

*A***Edit**menu from the menu bar and click

**Properties**.

** Note:** Be sure that only point

**is selected. If more than one object is selected, click the blank part of the drawing area, then click point**

*A***.**

*A*6.) In the *Properties dialog* box, click the *Label* tab and change the name of** A** to

**as shown in**

*P**Figure 3*and click the

**OK**button when you are done.

7.) Rename point ** B** to

**. You can display the**

*R**Edit Properties*dialog box by right-clicking point

**and choosing**

*B***Properties**from the pop-up menu.

8.) Select the *point tool, *and construct a point on the drawing pad not on line ** PR**.

9.) Display the name of the point. If you have followed correctly, the name of the point should be point ** C**, otherwise you rename it

**. (The names of the objects do not really matter much, but we will refer to them later using their names, so it is important that you rename them as instructed).**

*C*10.) We construct a line parallel to ** PR** and passing through point

**. To do this, select line**

*C***(click the line, not the points), select point**

*PR***, click the**

*C***Construct**menu and click

**Parallel line**from the list. Move points

**,**

*P***and**

*C***. What do you observe?**

*R*11.) We construct a point on the line parallel to ** PR**. To do this, click the line parallel to

**passing through**

*PR***, and click**

*C***Point on Parallel line**.

12.) Display the label of the fourth point (refer to number 4) and rename it to ** Q** (refer to number 6 or 7). Move point

**and point**

*Q***. What do you observe? What is the difference between point**

*C***and point**

*Q***?**

*C*13.) To construct the interior of triangle, select point ** P, Q** and

**(be sure that only the three points are selected), then click the**

*R***Construct**menu and click

**Triangle Interior**. Your diagram should look like the one shown in

*Figure 5*. Move point

**. What do you observe about the interior of the triangle?**

*Q***Part II: Exploring the Properties of the Dancing Triangle**

We have finished our construction. Now, we will observe what happens to the area of the triangle if we move point ** Q** along the line parallel to

**.**

*PR*14.) To display the area of the triangle, click the interior of the triangle, click the **Measure** menu, then click **Area. **Notice that a text containing the area of ** PQR** appeared at the left corner of your drawing area.

15.) Move point ** Q**. What do you observe? Can you think of the reason why your observation is such?

16.) The area of a triangle is the product of its base and its altitude. So, let us see what happens to the length of the base and the altitude when we move point ** Q**. To display the length of the base we just have to find the distance between

**and**

*P***. To do this, click points**

*R***and**

*P***, click the**

*R***Measure**menu and click

**Distance**.

17.) Move point ** Q**. What can you say about the measure of the base of the triangle?

18.) Next, we construct the altitude of triangle** PQR**. Since the altitude of the is perpendicular to its base, we must construct a segment perpendicular to

**and passing through**

*PR***. To do this, click line**

*Q***(not the points), select point**

*PR***, then click the**

*Q***Construct**menu and select

**Perpendicular Line**form the list.

19.) Next, we construct the intersection of * PR* and the line perpendicular to it. To do this, select line

**(not the points), click the line perpendicular to**

*PR***, then click the**

*PR***Construct**menu and click

**Intersection**form the list.

20.) Display the label of the intersection point. Its name should be ** D**.

21.) We hide the line ** QD, **but leaving points

**and**

*Q***. To do this, right click line**

*D***(not the points), then click**

*QD***Hide Perpendicular Line**s, then select the

*segment tool*, click point

*and click point*

**Q***.*

**D**22.) To turn altitude ** QD** to a dashed line, right click it and choose

**Dashed**from the pop-up menu.

23.) Our last step is to display the length of ** QD**. To do this, select segment

**(not the points), click the**

*QD***Measure**menu, then click

**Length**.

24.) Now move the points on the drawing. What do you observe?

25.) Move point ** Q** along the line. What do you observe?

26.) Explain why the area of the triangle is constant when you move point ** Q**.

In the Geometer’s Sketchpad Tutorial 3, we are going to learn how to construct graphs and sliders.