Tuesday, February 21, 2006

How To Make Homemade Ramen Soup


On the notion of congruence of triangles

Equality and consistency

The concept of congruence is related to the equal and it is expected that the learner knows it, either intuitive meaning from natural language or through use in arithmetic. It is customary to speak of congruence geometry rather than equality. For example, two segments are congruent if and only if they have the same measure, and the same is true for angles. But in the case of two triangles, the definition is more complicated because there is no measure (number) that defines a triangle.

triangle as a configuration of points and lines

As we know, there are different classifications of triangles that account for their diversity of form: according to the measure of their angles can be obtuse, rectangles, acutangula, in accordance the relationship of the measures of its sides can be equilateral, isosceles, scalene. That's why a pre-defined notion of congruence of triangles is the correspondence. This is because a triangle (and any polygon) is a configuration consisting of points and line segments (sides) that connect pairs of points.

Congruence triangles as intuitive notion and its formalization

Having discovered that two triangles are congruent (equal) should put their corresponding vertices. To say that the triangle ABC is in correspondence with IJK means that the correspondence between its vertices is AI, BJ and CK. And in this correspondence is implicit in the correspondence between the sides: AB-IJ, JK and BC-CA-KI. But it is also implicit correspondence between the angles: the angle at A is congruent to angle R, etc. (Note: not all texts follow this convention, that is, even when claiming "ABC is in correspondence with IJK "do not respect the above rules of implied correlation-a shame ... but what are you going to do.)

And when I say" discovered "I mean the view consistency cognizable by intuitive and informal methods, or perhaps rather, "sees." But once you "see" the consistency should be formalized. This is desirable because once the correspondence and consistency in the way explained above, it is not necessary to see the figure to raise equations or reasons, then the correspondence between vertices and sides are implicit in the correspondence between the triangles as already explained.

To see the need to search for consistency, that is, something (a sentence, a fact, ...) in the problem statement to suggest that consistency can be used for its solution. And to find it, once you seek it is convenient to use the intuitive definition: two triangles are congruent if they can be matched one on the other by rotations, translations and / or reflections. (The formal definition is: two triangles are congruent if, in the correspondence between their vertices, are equal to the corresponding sides and corresponding angles.) In a triangle congruence then have six pars, three sides and three angles. It is therefore very useful have criteria that tell us whether two triangles are congruent without having to verify the six equalities.

matching criteria as postulates

The criterion (principle) of consistency is perhaps the most basic criterion called LAL (side-angle-side) tells us that if, in a letter of triangles, two sides of one and the angle between them are equal to their corresponding elements in the other, then the two triangles are congruent. Some texts of formal geometry, the most in the logical sense, taking this approach as an axiom and show the remaining two, the ALA and the LLL. Other texts-most- postulated as true the three criteria. It is recommended then that the learner's take the three as postulates for if in any way is going to take a postulate ...

In the figure, the triangles ABC and AB'C 'are in correspondence. The second is the result of the first rotated 90 degrees. If the missing segment BC, however the distance between A and B would remain after the turn.

Instance of use (classical) of the LAL test

isosceles triangle theorem:

If a triangle is isosceles then its base angles are equal. (Note: it is customary to understand the basis, the third side, the first two are the ones who know the same.)


Warning: This instance of use is somewhat disconcerting when you first see it, so it asks reader's cognitive cooperation. (The confusion is perhaps due to the triangle is placed in correspondence with himself, which is not forbidden but because one thinks that this ban was implicit in the definition of consistency.)

The isosceles is sample can be called triangle ABC. But, crossing the vertices in the opposite direction can be called triangle BAC. Correspondence is valid for ABC-BAC.

Since the triangle is isosceles with CA = CB and BC = AC. Also, since it's the same triangle, the angle at C is identical to itself. There is therefore a correspondence LAL and the two triangles are congruent. But then the other elements put in correspondence are also equal. In particular the angle at A is equal to angle B.

second instance of use (also classic) the LAL test

In an isosceles triangle, the bisector of the vertex opposite the base divides the triangle into two congruent.


In the above figure draw the bisector of angle C and assume that intersects the side AB at M. By hypothesis and MCB ACM angles are equal. This suggests the CC correspondence. On the other hand, by definition, AC = CB. This correspondence suggests AB, and the other point common to the triangles formed by the bisector is M, which suggests the MM correspondence.

Thus, we test the correspondence ACM-BCM. We have, AC = BC and CM = CM, to be seen whether the angle formed by AC and BC is equal to BC and consisting of CM. But that is true because CM bisector. So we can use the LAL test to establish that the positions corresponding triangles are congruent. This congruence

well established are still several

Corollaries (for isosceles):

a) The bisector is also bisector (as AMC and BMC angles are equal and their sum is a plain, but also the corresponding sides AM and BM are equal, so that MC is perpendicular to the midpoint of the base)

b) The bisector is also medium (for AM = BM)

c) The bisector is also high (as AMC and angles BMC are straight) Final comments

can deduct the standard LAL LLL from applying the properties of an isosceles triangle, the triangles LAL is placed in correspondence as shown in the figure and ...

Since AB and AB = IJ = IK, we have the isosceles ABI and ACI. But then its base angles are equal. Adding, we find that the angles at A and R are equal and we are now able to apply the LAL test to ensure that the triangles ABC and IJK are congruent.

Say, finally, that the notion of congruence of triangles is very close to the foundations of Euclidean geometry. But the apprentice does not need to justify everything, especially near the foundation theorems. It is better, from the point of view of solving problems, to take the matching criteria as axioms and shamelessly use in solving problems. This allows you to move forward in its appropriation of theoretical tools without wasting time on formalities. Also be taken as equal angles formed by two parallel and a transversal. Of course it is desirable that some may see demonstrations of the basic theorems, but that can wait ... Meanwhile, to solve problems ... in VL

Sunday, February 19, 2006

What Does A Brazilian Shave Look Like

An elementary geometric problem

These open days at the University (of Tamaulipas) Saturday's workshop entitled "Science Workshop for young people." Responded to the call two high school teachers with 7 of its students, and a retired teacher.

The writer was in charge of the session with the intention to start developing the theme of "complex numbers and Euclidean geometry, a topic that I find very productive for solving geometric problems from an algebraic point of view.

The age of the participants (12 to 15) made me wonder, and better I ask you bring a problem of geometry that would like to address here with me? And his answer made me suspend the issue of complex and enter the matching of triangles, a common theme but has more potential than you might think to solve problems. Boys took out his notebook and I raised the

Problem 1:

In triangle ABC, with right angle at B, E and F are AC so that AB and AE = CF = CB. How long is the angle EBF?


I decided to accept the challenge of solving (help) this problem is elementary geometry, however, their fine detail. I started with a discussion about drawing the figure and evoke theoretical meaning from the data.

The condition of equal segments seems to suggest using congruence of triangles. But once you see a figure closer (about especially after drawing BF and BE) the hypothesis of congruence should be replaced by isosceles triangles.

It is therefore clear that the triangles ABE and BCF are isosceles. And once you are bringing to mind the concept of an isosceles triangle, with it comes the "base angles equal."

So far, the cognizable is the expectation that the idea of \u200b\u200bequal angles at the base will be of some use. And yes. Because it allows the implementation of the algebraic machinery: M = x + y, N = y + z ... And an elementary teoremita was not mentioned (the sum of angles of a triangle is 180 ...) comes to save the whole situation: M + N + y = 180.

Since, moreover, by data we know that x + y + z = 90 ... a bit of algebra leads us to the answer y = 45.

us comment, finally, it is extremely rewarding experience for a math teacher to have a teen audience interested. It is indeed an extraordinary experience because it is common to have a captive audience (and the worst is that the teacher is also captive) with all the implications it may have the adjective. And one of them is the indifference of the majority.

While it is true that everyday classroom tend to negotiations for peaceful coexistence teacher-student, it is also true that most of the time these covenants courtiers are not entirely satisfactory to the parties - at least for the teacher who is trying to satisfy two conflicting forces: the duty to be of quality education in response to a society that naively still waiting for the educational system and the educational reality has used students to make paper without any effort on your part. Neither good nor bad, it's just a fact of life in Mexico. (Does the fact that the OECD we stand at last in the long run might change the situation?)

JMD in VL greets ... and promises to post more often ... at least one problem was solved in the Saturday session of the workshop ...