Part 2: Parabola as loci of lines To see the pattern described in Part 1 distinctly, we would have to fold the paper dozens of times. With your neighbor, plan a geometric construction we could use to simulate the folding process as described in Part 1.Instead, we are going to simulate the activity using The Geometer's Sketchpad. Write down the key steps of the construction and share your ideas with the class. Carry out your construction using The Geometer's Sketchpad. There are several ways to simulate the activity in Part 1, using different features of The Geometer's Sketchpad. Specific steps have been recorded below for three different methods. Construct a segment from the focus to the point on the directrix (segment FG).Ĭonstruct two points, one as the focus (F) and one on the directrix (G).Construct a line (d) that will serve as the directrix.(Instructor Note: A Sketchpad file illustrating the simulation using the animate feature has been saved as parabola-animate.gsp.)Īll three constructions start in a similar manner.Construct the perpendicular bisector segment FG.Select the perpendicular bisector, and choose the Trace Perpendicular Line command under the Display menu.Select point G (the driver point) and the perpendicular bisector.Note: There is an advantage to the construction using the Locus feature over the others. (Instructor Note: A Sketchpad file illustrating the simulation using the locus of lines has been saved as parabola-locus.gsp.) Once you have constructed the sketch, you can easily manipulate the position of the focus and directrix and investigate the connection between them. What is the general shape formed by the loci of the perpendicular bisector? Drag the focus point to manipulate the loci.Conjecture what happens when the focus is below the directrix.What is the relationship between the location of the focus and directrix and the general shape formed by the loci of lines?.What is the relationship of each of the perpendicular bisectors to the parabola? 7 Part 3: Parabola as a loci of points Why does the construction tracing perpendicular bisectors of segment from the focus to the directrix produce a parabola? To be able to answer this question we will need to slightly modify our construction.In Part 2, we constructed a parabola by loci of lines. Since each of the lines were tangent to the parabola, we could not locate any specific points on the parabola. We would like to be able to construct only those points on the parabola. Re-simulate the activity in Part 2 to construct a parabola as a locus of points.To construct a parabola as a locus of points: (Instructor Note: A Sketchpad file illustrating the simulation using the locus of points has been saved as parabolalocuspts.gsp.) Construct a dashed line through point G that is perpendicular to the directrix.Construct the point of intersection of the dashed line and the perpendicular bisector of segment FG.Construct the locus of point P as G moves along the directrix.What is the general shape formed by the loci of point P? Drag the focus point to manipulate the loci. The Perpendicular Bisector Theorem explains that any point along the perpendicular bisector line we just create is equidistant to each end point of the original line segment (in this case line segment AB).What do you observe about the shape formed? St ep 7: Now we can mark the point of intersection created by these two intersecting arcs we just made and draw a perpendicular line using a straight edge going through Point B and we have created our perpendicular line! Perpendicular Bisector Theorem: Swing the compass above the line so it intersects with the arc we made in the previous step. St ep 6: Keeping that same length of the compass, go to the other side of our point, where the given line and semi-circle connect. Then swing our compass above line segment AC. Step 5: Next, open up the compass at any size and take the point of the compass to the intersection of our semi-circle and given line segment. Step 4: Place the compass end-point on Point B, and draw a semi-circle around our point, making sure to intersect the given line segment. Step 3: First, let’s open up our compass to any distance (something preferably short enough to fit around our point and on line segment AC). Step 2: Our goal is to make a perpendicular line going through point B that is given on our line segment AC. We are going to need a compass and a straightedge or ruler to complete our construction. Step 1: First, notice we are given line segment AC with point B, not in the middle, but along our line.
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