By:

Heather Bridges

Let's start off by just defining what quadrature means. The quadrature
of a plane figure is the construction of a square having the same area as
the plane figure using only a compass and straight edge. To begin with something
more basic, let's see if a rectangle is quadrable.

To start the students off walk them through the steps to squaring the rectangle. Then give the students time to go to
the computers (GSP) and construct the solution of their own.

Steps:

1. Construct any rectangle

2. Extend AC to the right and mark off distance from CD on AC so that CH
= CD.

3. Find the midpoint of AH.

4. Construct a circle with center M and radius AM.

5. At C construct a perpendicular to AC through the circle.

6. Make point E at intersection of line and circle.

7. This gives the side length of the square, so finish constructing the
square.

8. Test the sketch to see if was constructed correctly by measuring areas
and dragging points.

Why does this work? Proof

Now look at the quadrature of a triangle.

Steps to the quadrature of a triangle:

1. Find the midpoint of segment BD.

2. Draw a line parallel to AB (the base) through the midpoint.

3. Rotate the triangles formed, triangle CME and triangle CMF 180 degrees
or dras perpediculars at points A and B to form the rectangle.

4. Follow the steps above for the quadrature of a rectangle.

Why does this work? Proof

Now that everyone has an idea of what quadrable means let's move on to the
lune.

What is a lune?

A lune is a figure in a plane that is bounded by two circular arcs.

Examples:

Can we create a construct a square with the area of a given lune? That
is our goal.

First of all there are some things that we need in order to be able to do
this effectively.

Assignment: Go to the computers and use Geometer's Sketchpad to construct
5 semicircles of whatever size just as long as they are different. Then
inscribe a triangle in each one of the 5 semicircles. Classify each triangle
according to the angles

Hints: How do we construct a semicircle? Construct a circle. Construct the
diameter.

Construct a arc on the circle and hide one side of the circle.

How do you inscribe a triangle in a semicircle? Make the diameter one of
your sides

and put your point somewhere on the arc.

Sample Of Students Work:

What can we conclude from everyone's inscribed triangles in a semicircle?
A triangle in a semicircle is a right triangle. Click
here for a demonstration.

The next assignment is for the students to construct two circles. Measure
their areas and the length of their diameters. Then compare the ratio of
the areas of the circles to the ratio of their diameters squared. What now
can we conclude about the relationship between the areas of two semicircles
and their diameters ?

Explorations:

Because the ratio of areas of the circles equals the ratio of the squares
of the diameters then because the area of the semicircles is just half of
the area of the circle then

We are going to use these things to complete our original question:

1. The pythagorean theorem

2. An angle inscribed in a semicircle is right.

3. The areas of two semicircles are to each other as the squares of their
diameters.

(Next step is for the teacher to try to lead the students through forming
their own proof of how to find the area of the lune based on what the know.)

Leading questions and hopeful answers:

1. What can we say about the relationship between triangle ABC and ABG?

We can conclude that they are congruent using SAS because AB = CB = BG =
radius.

2. What can we say now about AC and AG? They are equal.

Now lets see what we can do with the Pythagorean Theorem.

3. No let's see if we can apply what we learned about 2 semicircles and
their diameters.

(Give the students time to explore independently.)

So the area of the semicircle AEC = 1/2 the area of the semicircle CAG.

4. Does anyone have any idea where to go from here?

The area of the quadrant (AFCB) = 1/2 the area of the semicircle CAG so

the area of the semicircle AEC = the area of the quadrant AFCB.

ACE + CDE

equals semicircle ACE = quadrant AFCB which equals

ACE + triangle ACB .

So all we have to do is subtract the area of section ACE and we are left
with

Area (lune ACEF) = Area (Triangle ACB).

Since we have previously discussed how to construct a square whose area
equals that of

a triangle then we have successfully compeleted what we set out to do.

Dunham, William. *Journey Through Genius: The Great Theorems of Mathematics.*
(1990): 1-20.

Return to Great Theorems Page