Explorations in Geometry

with the

Geometer’s Sketchpad

Exploration 7

    1. Construct an angle with two rays at point A. Create a point, P, interior to the angle. Now from an adjustable point, D, on one of the rays, draw another ray through P until it intersects the other ray From A. I labeled this point F. Create the triangle ADF and measure the Area and Perimeter.
    2. Move point D to find the line DF which will give the minimum area of triangle ADF. Move point P and alter the angle and repeat this exercise. Can you predict where to place the line for different locations of P and different angles. Describe the relationship of the figure when the area is a minimum.
    3. This may be a somewhat more difficult exploration… Repeat the operations above to find the line DF which minimizes the perimeter of triangle ADF. Describe the relationship of the figure when the perimeter is a minimum. (hint, there may be another line that makes this easier to see) This line is often called Philon's Line. You can find a solution to this problem by looking up Philon's line at MathWords and Other Words
    4. Can you find an algebraic relationship between the angle at A, the location of P and the minimum area or perimeter? It might help to create a coordinate system with A at the origin and the line AF along the x-axis. Then the slope of line AD would be the tangent of Angle A.
    5. What other geometric or algebraic relationships can you find in this figure?
    6. Write a brief paragraph explaining what you have found in this exploration. Remember to answer all the specific questions above.