In this assignment I discussed two problems that I thought were interesting. The first problem looked at the relationship of a triangle and another triangle whose sides have lengths equal to the medians of the first triangle. The second problem looked at constructing a parabola given a fixed point for the focus and a line for the directrix.
For the first part of the assignment, I looked at the relationship between a triangle and the triangle whose sides are the lengths of the medians of the original triangle. I first constructed a triangle and its medians.
Then, I construct a second triangle with the three sides having the lengths of the three medians from the original triangle. To do this, I first constructed a line parallel to AB at point D and a line parallel to GF at E. By doing this, I will use DE as one of the sides of the triangle. The two parallel lines that I am constructing will intersect. This must happen because of the Euclidean Parallel Postulate. The intersection point of these two lines will form two segments whose measures are equal to GF and AB.
I provided the measures of these segments just in case you are not convinced by my argument above. Although I provided these measures, to understand this problem, you must understand the argument made above.
The question that is being addressed is what relationship exists between the original triangle BGD and the new triangle ECD, with sides equal to the medians.
The relationship between these two triangles is that the area of triangle ECD is 3/4 the area of triangle BGD.
Does this hold for any triangle? The following link allows you to explore the relationship of the areas using GSP 4.0. By clicking on either B,G, or D, the three vertices of the original triangle, you can change the shape of the original triangle which also changes the shape of the new triangle that was constructed. Notice that although the areas of the triangles are always changing, the ratio between the two areas is always .75. Explore.
For the second part of the assignment, I looked at constructing a parabola from a fixed point and a line. To do this, construct a perpendicular from the point to the line. Then pick another point on your given line. Now, construct a segment between your given point and the point you just chose. The next step is to construct the perpendicular bisector of the segment just constructed. Construct a perpendicular line through the chosen point on the original line and through the original line. Mark the intersection with a point. Here is what your construction should look like.
Now, to get your parabola, you must trace the last point that you constructed, point G in my construction, and move the point that you chose on the given line, point C in my construction, along that given line. Here is the animation of this. To run the animation, click on the animate point box.
You can also see the parabola when you trace the line tangent to the parabola. This line is the perpendicular bisector you constructed in the first part. Trace that line and move the same point, point C for my construction, along the given line. To see the animation of this, click here and then click on the animate point box.