Assignment #5

Angle Bisectors of Triangles

1. Draw a triangle and name the vertices.

2. Create an angle bisector through the following steps.

a. Choose an angle within the triangle by selecting the appropriate vertices.

b. Go to the CONSTRUCT menu and select ANGLE BISECTOR.

c. Voila!

3. Create a point of intersection between the angle bisector and the side opposite the angle.

a. Select the angle bisector and the opposite side.

b. Go to the CONSTRUCT menu and select INTERSECTION.

c. What happens?

4. Create the segment between the vertex of the angle and the point of intersection of the angle bisector and the opposite side by selecting the vertices that define the segment. Then go to the CONSTRUCT menu and select SEGMENT.

5. Hide the ray that represents the angle bisector by selecting the ray (not the segment you just created). Then RIGHT CLICK and select HIDE BISECTOR.

6. Continue to create angle bisectors for all 3 vertices of your triangle

7. Make sure you create a point of intersection between the angle bisectors and the opposite sides, and hide the angle bisector. You should have a triangle with 3 segments that are angle bisectors.

8. Label all points of intersection. Make sure you do not label any edges…or else… (FYI: Edges are labeled with lower case letters.)

9. There are two ways to measure lengths.

a. One way is to highlight the side you want to measure. Go to the MEASURE menu and select LENGTH.

b. Another way is to select the endpoints of the segment. Go to the MEASURE menu and select DISTANCE.

c. Now measure the lengths of all of the segments in your picture.

10. Now it’s time to think (doh!). Make all the observations you can, and record your observations. Consider ratios of these measurements where appropriate. Change the shape of the figure to observe changes in the measurements and ratios.

11. Describe how you would measure the distance between a point and a line.

12. Measure the distance from the point of intersection of the three angle bisectors to each of the three sides of the triangle. The distance between the point and the side is the shortest distance between the point and the side. Select the point and the side you want to measure the distance to. Then go to the MEASURE menu and select DISTANCE.

13. What do you observe? Be sure to change the shape of the triangle to solidify your observations.

14. What is the effect of changing the shape of the triangle mathematically? In other words, if you could not use GSP, what would you have to do that would be equivalent to changing the shape of the triangle.

15. Now construct a perpendicular line from one of the sides through the point of intersection of the 3 angle bisectors. Select the point and a side. Then go to the CONSTRUCT menu and select perpendicular line.

16. Create a point of intersection where the perpendicular line meets the side, and hide the perpendicular line.

17. Create a circle centered at the point of intersection of the 3 angle bisectors such that the circle has radius equal to the distance from the point to one of the sides of the triangle.

18. What do you observe? How do you know? Record your finding and discuss them with your neighbors.

19. Try this entire process with a quadrilateral. Can you inscribe a circle in any quadrilateral? Why or why not?