GeoGebra Lab 7

Perpendicular Bisectors

1. Open a new GeoGebra file. Turn off the grid and the axes in the Graphics View.
2. With the polygon tool, draw an arbitrary (nothing special about it) \(\Delta ABC\).
3. Click inside the triangle with the angle tool to measure all three of its angles. Make sure that \(\Delta ABC\) is an acute triangle. If it is not, press escape and move the vertices until all three angles are acute.
4. Using the Midpoint tool, construct the midpoint of each side of the triangle. Right-click one of the midpoints, and in Object Properties, change the color of each midpoint.
5. Select the Perpendicular Line Tool and select each midpoint and side of the triangle to construct the perpendicular bisectors of each side.
6. Notice that the equations of three lines have appeared in the Algebra View to the left.
7. Construct the intersection point of all the perpendicular bisectors. Change the name of the point to ‘Circumcenter’.
8. Save this sketch as GeoGebra Perpendicular Bisector Lab on your computer.

Answer the following questions in a textbox on your graphics view:

1. Modify your triangle by moving the vertices. Observe what happens to the circumcenter. What happens to this point when the triangle is right?
2. What happens to this point when the triangle is obtuse?
3. What happens to this point when the triangle is acute?
4. The circumcenter also has another interesting property. Recall the property of perpendicular bisectors discussed in class. The intersection of the perpendicular bisectors then has that property for both segments. So, what do you think is true of the circumcenter?
5. Check your conjecture by selecting the circle tool. With the cursor click on the circumcenter and drag the mouse to one of the vertices of the triangle (it doesn’t matter which one – why not?). Describe the circle in relation to the triangle.