If you later want to change the length of the segment to some other fixed length, you can show the hidden objects and edit the parameter. Construct the midpoints of the three sides. #Construct by center and radius gsp5 how to#Ĭonstruct the feet of the altitudes of the triangle ABC.With a straight edge, construct a line on your piece of wax paper.Construct a point anywhere on the paper, except on the line, and label it F.Using your straightedge, construct the line segment FG.Ĭonstruct another point anywhere on the line, and label it G.Fold your paper so that the two points are concurrent, and deliberately crease it so that you can easily see the fold mark when the paper has been flattened out. Fold your wax paper so that point F falls anywhere along the directrix.Ĭonjecture any relationships you might see between the line segment FG and the folded line.Again, deliberately crease your wax paper so that you can easily see the fold. Continue this process approximately ten more times so that each time point F falls on a different location of the directrix. We are interested in the pattern of the creases that are formed when point F is folded along the directrix. Flatten out your paper and look for any geometric patterns. Make a conjecture about the relationship of the distance from the focus to the boundary, and the distance from the boundary to the directrix.Describe the boundary of the shape of the area containing the focus that is bounded by all of the fold lines. 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.What do you observe about the shape formed?
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