Quadratics

6.3 Folding Paper, Equidistance, and Reflections

This section assumes some familiarity with ideas from a standard geometry course.

 

We need one preliminary observation, which is its own folding activity:

Take a blank piece of paper and draw two points.

Q6_3_pic1

Holding the paper up to the light, make a fold in the paper that brings one of the points directly on top of the other. Make a sharp crease along this fold.

Q6_3_pic2

What does the straight line given by the crease represent? What is its relationship to the two initial points? If \(P\) is a point on the crease, what can you say about its distance from each of the two original points?

 

Answers: A little thought shows that if we fold a point \(A\) onto a point \(B\) the crease is a line that passes through the midpoint of the segment \( \overline{AB}\) and at to \(90\) degrees it. This line is called the perpendicular bisector of \( \overline{AB}\).

Q6_3_pic3

Any point \(P\) on the perpendicular bisector of \(\overline{AB}\) is equally distant from \(A\) and \(B\). This follows from Pythagoras’s theorem:

Q6_3_pic4

\(PA = \sqrt{PM^{2} + AM^{2}} = \sqrt{PM^{2} + BM^{2}} = PB\)

And one can show that any point NOT on this perpendicular bisecting line is not equidistant from \(A\) and \(B\). (Draw a perpendicular line from the point to \(\overline{AB}\). It misses the midpoint. Then Pythagoras theorem shows we do not have equidistance.)

 

Thus:

The set of points equidistant from \(A\) and \(B\) are precisely the points on the perpendicular bisector of \(A\) and \(B\)

AND

This is the crease mark that arises from folding \(A\) onto \(B\).

 

We are now ready for the big folding activity of interest.

THE FOLDING ACTIVITY

Take a blank piece of paper and draw on it a straight line and a point \(P\) not on that line. Use a thick marker to make the point conspicuous.

Q6_3_pic5

Holding the paper up to the light, fold the paper in such a way that the point lands somewhere on the line. Make a sharp crease.

Unfold the paper and then make a second that takes the point \(P\) to a different place on the line. Make a second sharp crease.

Repeat this action at least FIFTY more times, making another fifty creases in the paper given by taking the point \(P\) to different locations on the line.

The 52 creases you have outline the shape of an interesting curve.

With a pencil draw along the creases to outline the curve you see.

Select a point on the curve you just outlined. With a ruler, measure the distance of this point from the original point \(P\) and measure its distance from the original line you drew. What do you notice? Repeat with three more points on the curve.

 

When completing this activity, you should see a U-shaped curve. The question, of course, is this curve the shape of a quadratic/parabola?

The COMPANION GUIDE to this QUADRATICS course answers this question for you!

 

EQUIDISTANCE BETWEEN A POINT AND A LINE

The previous folding activity asked as to measure distances between a point and line. Recall that we measure the distance of a point from a line via the length of a perpendicular line segment from the point to the line:

Q6_3_pic6

Suppose we are given a line \(L\) and a point \(F\) not on that line. Then the set of all points that are equidistant from the point and from the line (see the diagram below) give a U-shaped curve.
The point \(F\) is called the focus of this special curve and the line \(L\) is called its directrix.

Q6_3_pic7

This curve looks U-shaped. Is it quadratic?

The COMPANION GUIDE to this QUADRATICS course answers this question for you!

 

REFLECTION PROPERTY OF THE PARABOLA

Parabolas have the astounding reflection property that all incoming parallel rays of light, perpendicular to the directrix, are directed to the focus of the parabola.

Q6_3_pic8

Satellite dishes and reflecting telescopes use dishes with parabolic cross-sections so as to focus parallel rays of light to a fixed point, and conversely, search-light reflectors and automobile headlight reflectors, for example, are parabolic: all rays from a bulb positioned at the focus are reflected parallel to the axis of the parabola.
We need to prove this reflection property true!

 

The COMPANION GUIDE to this QUADRATICS course provides all proofs to all claims made in this section!

 

 

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