Example 4. Constructing a right triangle (with an introduction to perpendicular lines and points on a line).
To illustrate the use of perpendicular lines, we construct a right triangle first. Next, we construct an isosceles right triangle using what we know about perpendicular lines. Remember that the main property of a right triangle is to have a right angle (which means that two of the sides are perpendicular to each other). Open a new sketch and construct a segment AB. Now we construct a perpendicular segment to this segment. To do so, we will construct a perpendicular line to the segment. After that, we will construct a segment on this line. To construct a perpendicular line to the segment, we need to select the segment and a point through which the perpendicular line passes. Since we want the perpendicular line to go through one of the endpoints, use the SELECTION ARROW TOOL to select both the segment AB and, say, point B. Once you have selected both objects, this is indicated on the left bottom corner of the sketch (in version 4 only). Open the Construct menu and select Perpendicular line. See Figure 11.
Figure 11
Once you select this command you should read (in version 4) the comment on the bottom left corner of the sketch -- it indicates the construction of a perpendicular line to a segment and through a point. This line appears selected (indicated by a thicker pink line). This selection allows us to perform other constructions on this line. One of the constructions that we need is a point on this line, which will serve as the third vertex of the triangle. To construct the third vertex, open the Construct menu and select the Point On Perpendicular Line command. Notice that the comment in the left corner indicates that the point is a random point on the line. This has some advantages, as we will see. The third vertex is on the line and it may be on either side of the vertex B. Since it is selected (see bottom left corner), you can click on it and hold the mouse to drag it. Place it wherever you like. See Figure 12.
Figure 12
We will hide the line now. Click on any spot on the screen to deselect the point and then select the perpendicular line. Once it is selected, open the Display menu (Figure 7) and select the Hide Perpendicular Line command. You are left with segment AB and a point, which is the third vertex of the right triangle. Label the point C and construct segments AC and BC. By selecting all three points, and using the Segments command in the Construct menu, the software constructs the polygon with three vertices. See Figure 13.
Figure 13
Now, we are ready to take the "Drag" test. Select the SELECTION ARROW TOOL and click on any part of the screen. This will deselect any objects. Select any vertex of the triangle. If you select vertex B and drag it, you will notice that when it moves, vertex C also moves but remains on a segment (namely BC), which is perpendicular to AB. This is the same as to say that the construction passes the drag test. Notice also that by dragging vertex B, you can rotate the triangle around vertex A. Similar situations happen had you selected vertex A instead. You will also notice that both sides, AC and BC, also change in length. This is because when we constructed a right angle and vertex C, the other two angles of the triangle are determined. So when dragging vertex A (or vertex B) the measures of the angles are also preserved. One could also drag vertex C. In this case, deselect point B by clicking on any point on the screen off the triangle. Now, click on vertex C and hold the mouse button to drag it. As you will notice there are not many ways to drag it. It only moves along a path. This path is the perpendicular line that is hidden. Again, by dragging point C we still have a right triangle and so the construction passes the drag test.