Assignment 6

Parabolas – GSP

By

Michelle Nichols

A parabola is the set of all points equidistant from a line, called the directrix, and a fixed point, called the focus.

Let’s consider the construction of a parabola in GSP, starting with a line and point not on the line…

1.     Construct a line (the directrix).

2.     Construct a point somewhere above or below the line.  Label this point F (focus).

3.     Connect the point and the line with a segment.

4.     Construct the midpoint of this line segment, and then construct the perpendicular line through this point.

5.     Select the point on the directrix, and the line, and construct a perpendicular line through this point.

6.     Construct the intersection of this perpendicular line and the previous perpendicular.

7.     Select the point on the line and then this intersection point, and construct the locus.

8.     This final step creates our Parabola.

We can also trace our point on the parabola, then move the point on the Directrix to obtain a similar picture (the same parabola).

How do we know this is a Parabola?  Using our construction and the definition of parabolas we know that this not only looks like a parabola, it is!  Within this construction are triangles that will help with the proof.  If we name each of our points, we can connect segments and prove the construction.  See the adjusted construction:

And the Proof:

·        By construction, |FM| ≈ |DM|.  Point M is the midpoint between F and D.

·        Angle DMT ≈ Angle FMT.  A perpendicular was constructed through M to create these 2 angles, therefore making both of them right angles.

• |MT| |MT|, by the Reflexive Property of Congruence, which says any figure is congruent to itself.
• So, ▲FMT ▲DMT, by SAS.
• We now know that |FT| |DT|, by CPCTC.
• So our construction works – The distance from the Focus to the Point on the Parabola is equal to the distance from the Directrix to the point on the Parabola.