Class Assignment 6
Name ___________________
Conics
Constructions using Geometer’s Sketckpad
© James J. Woeppel, 2000
(see note on last page)
I
Drawing an ellipse as described on page 52 of Davis using Geometer’s
Sketckpad:
Using the Point Tool, construct two point (that are approximately on a horizontal
line and not too close to the edge of the pad). Label them A and B (going from
left to right); these will be the foci of the ellipse. Choose the Circle Tool and
click point B dragging to form a large circle containing point A. (If part of the
circle goes off the edge of the pad, it will not matter.) With the Point Tool, click
on the circle above (or below) point A. Label this point C. Draw the radius BC
and the line segment AC with the Line (Segment) Tool. Making sure that
line AC is selected, find the midpoint by clicking Midpoint on the Construct
menu. Label this point M. After making sure that the point M selected. Select
the line AC with the Selection Arrow Tool. Construct the perpendicular to
AC at M with the Perpendicular Line command on the Construct menu.
Making sure the perpendicular is still selected, select the line BC with the
Selection Arrow Tool, and find the point of intersection by clicking Intersection
on the Construct menu. Label this point P; it is on the ellipse.
We need the lengths AP and BP. With the Line (Segment ) Tool, draw the line
segment AP and click Length on the Measure menu. Then draw the line
HJJJG
segment BP and click Length on the Measure menu. (The line BC is already
there but we the segment BP too.) Calculate the sum of these lengths by
choosing Calculate on the Measure menu. Then click m AP, hit the + key, click
mBP, then click OK.
We use the Locus command on the Construct menu to draw the ellipse. Click a
blankspace with the Selection Arrow Tool to deselect everything. Select the
point P and the point C (in that order) using the Selection Arrow Tool. Then
click Locus on the Construct menu. The ellipse with foci A and B is drawn.
The construction works because the right triangles ªAMP and ªCMP are
congruent. Thus AP and CP have the same length. The line segment CP is
part of the radius BC. The other part is BP. Therefore, the sum of the lengths of
AP and BP is the radius BC which is constant. You can test this by using the
Selection Arrow Tool to move the point C on the circle. The sum remains
constant.
2
Draw the line segment AB with the Line Segment Tool and find the midpoint
using the Midpoint command on the Construction menu. Label the point O and
draw the line segment OB measuring with the Length command on the
Measure menu. Click a blank space (to deselect everything). Draw a line
through points A and B by selecting them and clicking the Line command on the
Construct menu. Click a blank space (to deselect everything). Click the ellipse
to select it and then click Point on Locus on the Construct menu. Move this
point with the Selection Arrow Tool to the line through points A and B and label
it R. Draw the line segment OR with the Line Segment Tool and the measure it.
Click Calculate on the Measure menu, and then click mOB, click ÷ on the
calculator, click mOR , and finally click OK on the calculator. Click Calculate
(again) on the Measure menu. Click Values on the calculator and then
mOB mOR on the menu that appears. Finally click OK. Right click (click with
the right mouse button) the dialog box that just appeared. Click Properties on the
menu and then the Label tab. Fill in Label box with eccentricity and click OK.
You can hide the objects other than the ellipse and its radii by selecting them and
clicking Hide Objects on the Display menu. That way no one can tell how you
did it; just like a mathematician.
3
II
A hyperbola is the set of all points in the plane the difference of whose distances
from two fixed points (the foci) in the plane is a positive constant. The
construction of a hyperbola is very similar to the construction of an ellipse above.
Using the Point Tool, construct two point (that are approximately on a horizontal
line, not too far apart, and not too close to the edge of the pad). Label them A and
B (going from left to right); these will be the foci of the hyperbola. Choose the
Circle Tool and click point B dragging to form a circle not containing point A.
(It should almost contain point A.) With the Point Tool, click on the circle above
(or below) point A. Label this point C. Choose the Line Tool by clicking on the
Line (Segment) Tool and while continuing to hold the mouse button sliding to
HJJJG
left choose the Line Tool (the double ended arrow). Draw a line BC through
points B and C with the Line Tool by clicking on point B and dragging the line to
point C holding the mouse button. Draw the line segment AC with the Line
(Segment ) Tool. (First, you need to switch back to the Line (Segment)Tool.)
Making sure that AC is selected, find the midpoint by clicking Midpoint on the
Construct menu. Label this point M. After making sure the point M is selected
and select the line segment AC with the Selection Arrow Tool, construct the
perpendicular to AC at M with the Perpendicular Line on the Construct menu.
HJJJG
Making sure the perpendicular is still selected, select the line BC with the
Selection Arrow Tool, and find the point of intersection by clicking Intersection
on the Construct menu. Label this point P; it is on the hyperbola.
We need the lengths AP and BP. With the Line (Segment ) Tool, draw the line
segment AP and click Length on the Measure menu. Then draw the line
HJJJG
segment BP and click Length on the Measure menu. (The line BC is already
there but we the segment BP too.) Calculate the difference of these lengths by
choosing Calculate on the Measure menu. Then click m AP, hit the - key, click
mBP, then click OK. Click Calculate on the Measure menu. Click Functions,
and then click abs on the menu that appears. Click the m AP − m BP; this
transfers the formula to between the ( ). Finally, click OK.
We use the Locus command on the Construct menu to draw the hyperbola.
Select the point P and the point C (in that order) using the Selection Arrow Tool.
Then click Locus on the Construct menu. The hyperbola with foci A and B is
drawn
The construction works because the right triangles ªAMP and ªCMP are
congruent. Thus AP and CP have the same length. The radius BC is part of the
line segment CP. The other part is BP. Therefore; the difference of the lengths of
AP and BP is the radius BC which is constant. You can test this by using the
Selection Arrow Tool to move the point C on the circle. The difference remains
constant.
4
Draw the line segment AB with the Line Segment Tool and find the midpoint
using the Midpoint command on the Construction menu. Label the point O and
draw the line segment OB measuring with the Length command on the
Measure menu. Click a blank space (to deselect everything). Draw a line
through points A and B by selecting them and clicking the Line command on the
Construct menu. Click a blank space (to deselect everything). Click the
hyperbola to select it and then click Point on Locus on the Construct menu.
Move this point with the Selection Arrow Tool to the line through points A and
B and label it R. Draw the line segment OR with the Line Segment Tool and
the measure it. Click Calculate on the Measure menu, and then click
mOB, click ÷ on the calculator, click mOR , and finally click OK on the
calculator. Click Calculate (again) on the Measure menu. Click Values on the
calculator and then mOB mOR on the menu that appears. Finally click OK.
Right click (click with the right mouse button) the dialog box that just appeared.
Click Properties on the menu and then the Label tab. Fill in Label box with
eccentricity and click OK.
III
A parabola is the set of points that are equidistant from a point (called the focus)
and a line (called the directrix).
Using the Point Tool, construct a point in the upper middle of pad. Label it F
(the focus). Some distance below the point F, construct a horizontal line with
the Line Tool. With the Point Tool, construct a point not below F on the
horizontal line; label it C. With the Line (Segment) Tool, draw the line segment
FC use the Midpoint on the Construct menu to find the midpoint and label it M.
Making sure the point M is still selected, select the line segment FC and construct
a perpendicular at M with the Perpendicular Line command on the Construct
menu. Now, click a blank space with the Selection Arrow Tool to deselect
everything. Also, using the Perpendicular Line command construct a
perpendicular to the horizontal line at point C. Making sure this perpendicular is
still selected and selecting the perpendicular at point M with the Selection Arrow
Tool, find the point of intersection of the two perpendiculars with Intersection on
the Construct menu. Label this point P; it is on the parabola.
We use the Locus command on the Construct menu to draw the parabola. Click
a blank space again with the Selection Arrow Tool to deselect everything.
Select the point P and the point C (in that order) using the Selection Arrow Tool.
Then click Locus on the Construct menu. The parabola with focus F is drawn.
5
Draw the line segment FP, measure it with the Length command on the
Measure menu. Also, draw the line segment CP and measure it. The construction
works because the right triangles +FMP and +CMP are congruent.
Thus FP, and CP have the same length. Move point C along the horizontal line to
see this.
Click Calculate on the Measure menu. Click m FP, then click ÷, and then
click mCP. Finally, click OK. Click Calculate again and use Values to recall
this quantity and then click OK. Label this quantity eccentricity using the
properties on the menu obtained by right clicking the box containing m FP mCP .
IV
Eccentricity of Conics. The eccentricity (e) of an ellipse or a hyperbola is c/a
where c is the distance from the center to a focus and a is the distance from the
center to a vertex. The eccentricity of a parabola is 1.
Open Geometer’s Sketchpad. On the File menu click Open and double click
Samples. Then double click Sketches; next double click Conics. Click Unified
Conics.gsp and then click OK
The author has created several Action Buttons; they can be activated by clicking
them. His definition for conics, which appears in the upper left hand corner, is
the same for all conics.
If you drag the point in the middle of the sketch that is labeled, E the eccentricity
of the conic will chance. Click the red Action Button “█Ellipse” (this moves the
point E to the left of the center of line segment FA ); the eccentricity is positive
but less than 1 giving an ellipse. If the point is dragged to the center of the line
segment (or click the red Action Button “█Parabola”); the eccentricity is 1
giving a parabola. If the point is dragged to the right of the center of the line
segment FA (or click the red Action Button “█Hyperbola”), the eccentricity is
greater than 1 giving a hyperbola.
Clicking on the blue Action Button “█Distances,” gives a definition of
eccentricity as the quotient of the lengths of two line segments, which is for all
conics (ellipses, parabolas and hyperbolas). The Action Button █Construction
gives the construction. (There are two points labeled S!)
© note: This material may be used by anyone. If it is reproduced, written permission must be granted by
the author: James J. Woeppel, Indiana University Southeast, New Albany, IN 47150. © James J. Woeppel,
2000