The Conics
By: Diana Brown
Day Four:
Ellipse Introduction
Construction of an
ellipse with wax paper
Start out with a piece
of uncrumbled wax paper about the size of half a sheet of standard notebook paper
(8.5 x 11). Draw a circle in the center of the wax paper without writing
off the paper. Then draw a point anywhere inside the circle excluding on
the circle and the center. The next step is to fold up the circle so it
touches the point inside. When this portion of the circle is aligned with
the point, crease the paper and fold it accordingly. Choose another part
of the circle and align this with the point, creasing the wax paper.
Repeat this step several times until the majority of the circle has touched the
point inside or until an ellipse is visible.
Construction of an
ellipse on Geometers Sketchpad
Draw a circle
Place a point anywhere inside the circle
(point A) and one on the circle (Point B)
Construct a segment from point A to B and
construct line segment AB’s midpoint (Point C)
Construct the perpendicular bisector of line
segment AB and while this line is still selected, choose Trace Line from the
Display menu.
To simulate the process of folding, point B and
under Display click animate point.
Watch the ellipse form!
Proof
Statement: The sum of the distances from two points
(the foci) inside an ellipse to any point on the ellipse is constant.
Proof: The foci on our sketch are
the points A (the center of the circle) and B (the point inside the
circle). We will again use Geometer's Sketchpad to visualize our
proof. Construct a line segment from A to C and place a point of
intersection E where this new line segment intersects the perpendicular line
passing through point D. This point E traces out the ellipse.
Construct another line segment between points E and B. You should now
have a triangle BCE with a perpendicular bisector ED. Because ED is the
perpendicular bisector, BD is congruent to DC and the angles BDE and CDE are
both 90 degrees. Of course, DE is congruent to itself, and thus we have
two triangles with two congruent sides with an included congruent angle.
By SAS, the triangles are congruent and therefore EB is congruent to EC.
The radius of the circle is AC = AE+EC = AE+EB. So since the radius is
constant, the sum AE+EB is always constant.
Ellipses in
real life:
Example 1) A portion of the white house lawn is called The Ellipse. It is 1060 feet long and 890 feet wide.
Example 2) The orbits of the planets create an ellipse.
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Five (Ellipses)