**Paper-folding construction**

- As we did with the construction of the parabola and the ellipse, we're going to make a hyperbola using wax paper.

2. Next, mark a point outside the circle. It really doesn't
matter where this point is, but it might be easier if it's closer to the
circle. This point is the **focus**.

3. Now, **fold** the paper so that the **circle **coincides
(lies directly on top of) the focus. Make a sharp crease of the fold.

4. Do this numerous times. Start at one point on the circle and work your way around the circle.

5. What do you notice happening? Is anything appearing on your paper.

**Geometer's Sketchpad Construction**

- Let's try to repeat this process using Geometer's Sketchpad.

2. Near the center of the screen, construct a **circle** with
a diameter of about 2-3 inches.

3. Create a point **(focus)** outside the circle.

4. **Select** the **circle** and **construct** a **Point
on Object.** (This point is movable along the circumference of
the circle. The other point that you see on the circle defines the
circle, and if you try to move it, the entire circle will move.)

5. When we created the parabola and ellipse, the next step was
to create a perpendicular bisector of the segment connecting the **focus**
and the
**sliding point**. Do the same for the construction of
a hyperbola. You can change to color of this line to aid in visualization.
This line is the fold line from the wax paper construction of an ellipse.

6. Now, we want to simulate the process of folding. Select
the perpendicular bisector and under **Display**, choose the option
**Trace
line.**

7. Next, select the **sliding point** and the **circle**.
Under **Edit**, choose **Action Button** and the type **Animation.**

8. Once the animation button is created, double-click it and watch.

It looks like there is a hyperbola on the screen now, but how do we know that what we have constructed is really a hyperbola?

Let's try to prove that what we have constructed is a hyperbola:

Under **Edit**, select **Undo** to remove all of the trace lines
from the animation.

What is the definition of a hyperbola?

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Let's go about this proof in a manner similar to what we did for the parabola and for the ellipse. How do we determine the distance from a point to a circle?

(**Hint:** it's perpendicular distance)

9. Construct a **line **that is perpendicular to the circle
by **constructing** a **line** through the **sliding point** on
the circle and the point that defines the center of the circle. This
line is perpendicular to the circle at this specific point.

10. Notice that this line crosses the fold line. Select
both lines and **construct** the **point of intersection**.

11. Next, **construct** the **segment** connecting the **new
point** to the **focus**.

- To prove that point F in the diagram would actually "carve" out a hyperbola, we need to show that the difference of the measurements of FC and FA = a constant (i.e. |FC-FA| = a constant).

13. The measurements should be in the upper lefthand corner at
this point. Select both measurements and under **Measure**, select
**Calculate**.
Under **Functions**, select **abs[** to get the absolute value of
the difference. Then, under **Values**, you can select the distance
of FC and FA. Subtract one value from the other, select **)**,
and click OK. The difference of the two distances should now be displayed
in the upper lefthand corner.

14. Select the fold line and turn off the trace function under
**Display**.
Select the point that defines the hyperbola (point F above) and under **Display**,
choose **Trace Point.**

Select the segment **CD, midpoint E, the perpendicular line, and the
fold line** and under **Display**, choose **Hide objects.**

Perform the animation. What do you notice? As the animation
runs, keep watching the distance measures from #13. What is significant
about this calculation? Does this help you prove that what you constructed
is truly a hyperbola?