Tag Archives: congruence

Common sense versus logic and math: Congruence again

I thought I would write a computer routine to check if two figures were congruent by the CCSS definition (rigid motions). One day I will post it.

The most important thing was to be specific as to what is a geometrical figure. You can read the CCSS document from front to back, back to front, upside down and more, but NO DEFINITION of a geometrical figure. For the computer program I decided that a geometrical figure was simply a set of points. My diagram may show some of them joined, but any two points describe a line segment (or a line). So a line segment “exists” for any pair of points.

The question is “Are the two figures shown below congruent or not?

congruent or not

I rest my case…..

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Conjecture……….true?……..false?………undecided?…..

conjecture jpeg1

and now for the dictionary

conjecture jpeg2

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Congruence. Transformations. Definitions. Unnecessary.

Once again I take the Common Core to task.

This time their obsession with transformations.

There is nothing wrong with learning about transformations but it is silly to attempt a definition of congruence in terms of transformations, as we will see. Rigid transformations of the plane (translations and rotations) preserve lengths and angles, and so a rigid transformation of a figure gives a new figure which is “the same”. Also, and ignored by CCSS is the vital fact that such a transformation can take an image to ANY position in the plane, where position is identified as “point + direction”.

So this is what is written:

High school geometry: Congruence
Preamble
The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here
assumed to preserve distance and angles (and therefore shapes generally).
In the approach taken here, two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one onto the other. This is the principle of superposition. (me – This last is garbage – see at end)

So here goes, an attempt to see if line segment FQ is congruent to BA

congruent line segs rigid transf

We slide BA in the direction B to F until B coincides with F – this is our translation.  A is now at the point K

We then rotate BA around point F until it is on top of  FQ, it is now in position FL. This is all possible as the position of the (simple) figure FQ must be known.

Have we carried BA onto FQ ? Not necessarily. It depends on where point Q is. The only way that L and Q can be the same point is a) if we know the length of segment FQ and b) this length is equal to the length of FL, which is the length of BA.

Therefore the only way that BA can be moved and placed on FQ is if they are the same length.

What have we done?  We have shown that if BA and FQ (or any two line segments) have the same length then there is a rigid transformation carrying one to the other. Hence by the definition of “congruent” BA and FQ are congruent.

In simple words, If BA and FQ have the same length then they are congruent, so simplicity (a highly desirable aim in math) is better served by the simpler definition:

TWO LINE SEGMENTS ARE DEFINED TO BE CONGRUENT IF THEY HAVE THE SAME LENGTH (AND not OTHERWISE)

The dragging in of transformations really muddies the water.
Here is part of the High School geometry detail

Understand congruence in terms of rigid motions
6. …..; given two figures, use the definition of congruence in terms of rigid motions
to decide if they are congruent.

and here is some dictionary stuff on superposition

http://www.thefreedictionary.com/superposition
 2.  The principle by which the description of the state of a physical system can be broken down into descriptions that are themselves possible states of the system. For example, harmonic motion, as of a violin string, can be analyzed as the sum of harmonic frequencies or harmonics, each of which is itself a kind of harmonic motion; harmonic motion is therefore a superposition of individual harmonics.
3. superposition – (geometry) the placement of one object ideally in the position of another one in order to show that the two coincide
locating, positioning, emplacement, location, placement, position – the act of putting something in a certain place
 4. superposition – the placement of one thing on top of another
locating, positioning, emplacement, location, placement, position – the act of putting something in a certain place

There is no Principle of Superposition in geometry!

 

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