Explorations in Geometry

with the

Geometer’s Sketchpad

Exploration 8

    1. Construct a triangle ABC and the angle Bisector of the angle at B.
    2. Measure the sides of the triangle and the two pieces formed on side AC by the angle bisector. Can you find a relationship between the measurements. Is it true for all shapes of triangles?
    3. Now measure the length of the angle biscetor segment itself. Can you find a relationship between this length and the measurements above? If you can, you've discovered a special case of Stewart's Thm.
    4. Find the area of the two sub-triangles ABD and BCD. Can you see a relationship between their sizes.
    5. Measure the perpendicular distance from points C and D to the line AB. What do you observe. Why might you have expected this relationship?
    6. Create the circle circumscribing ABC. Find the point where the angle bisector crosses the circle. Measure the length of the arc from this point to C and to A. What do you observe? Can you PROVE that it is always true?
    7. What other geometric or algebraic relationships can you find in this figure?
    8. Write a brief paragraph explaining what you have found in this exploration. Remember to answer all the specific questions above.