The purpose of this communication is to generalize the theorem of Pythagoras using the corresponding area formulas for different geometric figures used in experience; the aim is to look at the possibility of Demosthenes this relationship using different geometric figures squared, showing how calculators can be used to explore the situation and give account of the difficulties that students with geometric concepts.

1. Generalization of the Pythagoras Theorem
Eugenio Therán Palacio
Rector Institución Educativa Sabaneta – San Juan de Betulia, Colombia.
eugeniotheran@gmail.com
Introduction
To approximate generalization of the Pythagoras Theorem wing mean some
different point of views. One of them is the suggest from the approach
trigonometry. To assume with acute-angled and obtuse triangles that result in the
so-called law of cosines. Another perspective can be seen from the theory of
numbers, considering an expression of the form 𝑥 𝑛
+ 𝑦 𝑛
= 𝑧 𝑛
, which it is related
to Fermat's Last Theorem, which owes its name has since been one of the
theorems mentioned in the writings of Fermat1
.
Another perspective from which you can look at the widespread Theorem of
Pythagoras geometry, played in two-dimensional point space as the relationship
that not only meets constructing tables on each of the sides of a right triangle but
with any kind of polygon; 3d space, understood as an equation that relates the
inner diagonal of a cuboid with an expression of the form (D=a+b+d) (a, b and d
are the sides of parallelepiped and D is the inner diagonal).
The fourth perspective is technology. Mediated dynamic generation. From the latter
the present research experience considering the possibility of demonstrating the
Pythagorean relationship using different geometric figures squared through the use
and management of graphing calculator is inscribed. (TI92plus; CTI92P).
The purpose of this communication is to generalize the theorem of Pythagoras
using the corresponding area formulas for different geometric figures used in
experience; the aim is to look at the possibility of Demosthenes this relationship
using different geometric figures squared, showing how calculators can be used to
explore the situation and give account of the difficulties that students with
geometric concepts.
Methodological development of the classroom experience for the exploration of the
Pythagoras Theorem was undertaken with the tenth grades of School Normal
Superior School Corozal (I.E.E.N.S.C) Sucre – Colombia, taking a sample of 12
1
This theorem states do not exist three integers x,y and z verify the following equality for Natural n
greater than 2: x + y = z. in 1995 the English Andrew Willes proved Fermat's theorem. After eight
years devoted exclusively to it.

2. students. Systematizing experience gave through observation and analysis of the
records included 3 students - Maria and Carlos –alexander for a more serious and
accurate results, Mayra Alexander and Carlos- students for a more serious and
accurate results.
Development
The reasons for making this experience are focused on the traditional teaching of
the Pythagorean relationship through a formula without considering the possibilities
of analyzing it from the point of view of variation: recording the data in a table
regardless of any verification of the relationship with other regular polygons; and
making use of instrumental mediation with CTI92P. the experience was made in
eight working sessions of two hours each. In the form of workshop. In which three
phases are worked: the first. Individual work by transit exploration-production-
construction second group work where agreements and arguments to try to answer
the questions raised in the workshop were presented. And the third. Socialization
of work involved the circulation of knowledge built through a pooling overall. In
which each group presented its findings and recommendations as well as the
difficulties and progress.
The activity began presenting guidelines for the construction of a triangle and a
rectangle as example for future buildings where the application of the Pythagorean
theorem with different regular polygons be displayed. (Figure 1).
The following questions were raised: ¿What about the triangle ABC when the T
point is on the RS segment moves? ¿What properties are invariants in the triangle
ABC to move the point T on the RS segment?
Subsequently each of the sides of the triangle ABC is constructed square (Figure
2).

3. ¿What is the relationship between the areas of the large square and the sum of the
areas of least squares?
Deletes the squares of the previous figure and the triangle rectangle ABC with the
right angle A (legs and hypotenuse) equilateral triangle are constructed and the
ratio between the areas of the triangles (Figure 3) from the following states
Question.
¿What relationship can be established between the areas of the equilateral
triangles?
¿What happens to the area of the largest equilateral triangle and minor amounts of
equilateral triangles when we move the point T on the RS segment?
This process is repeated with pentagons, hexagons and semicircles as you can
see in the figures 4, 5 and 6, asking questions that have Consistency with said
earlier:
After a table2
built starting in the area of the square on the hypotenuse and the
sum of the areas of the squares smaller is constructed. in a ABC triangle, rectangle
in A.
Observing the table studying the case with the corresponding values with the area
of greatest box shown in the first column and corresponding to the sum of the
areas of least squares appearing in the second column values.( table 1).
2
Using the Data Editor (DATA MATRIZ EDITOR) the graphing calculator, the input variables are defined
and then the point T of the RS segment is encouraged.

4. Then the graph3
that represents the area of the larger square and the sum of the
areas of least squares is built when a leg varies its length. (Figure 7).
According to the graph. ¿How they are related variables?, ¿You could algebraically
establish this relationship?, ¿How would you do it? What is the equation that
relates the variables area of the larger square sum of the areas of the two smaller
squares?, ¿What from that equation you can conclude? What is the value of the
slope?, ¿What is the value of the intercept with the Y axis? How these values are
interpreted?
Additionally you can ask other questions to expand exploration as the following:
¿how will relate the lengths of the hypotenuse and leg lengths and variable area of
the large square?, ¿How they are related variable length leg and the sum and the
areas of small squares?, ¿Is it possible that the Pythagorean theorem is fulfilled
constructing such polygons on each of its sides?
3
Using the Graph Editor (graph) starting of the linear regression equation.

5. Some results
In the first question from the teacher ¿what about the triangle ABC when the T
point being on the RS segment moves? According to the answer given by Myara
some confusion is detected by referring segments AC and CB as triangles as if
they were perhaps: this is caused by the haste with question is answered. When
the AB segment states are unchanged its assessment is correct but the justification
is connected with the variation of the AC segment. Here refers to Hicks segments
cathetus AC.
Alexander interprets as a segment thereof variability appreciated however
curiously it refers to the segments AB and CD as being points.
The next question ¿what properties are invariant triangle ABC by moving the
point T on the segment RS? Mayra follows invariance ab Hick and the sum of the
internal angles of the triangle ABC; alexander persists to notice the ab segment as
a point. Carlos observes another invariant property that had not detected his
companions. such as the permanence of the right angle A.
Asked about the relationship that existed between the areas of the squares on the
sides of the triangle ABC, Mayra quickly identifies the essential characteristics of
the Pythagorean relationship: also observed invariance of the square whose side
the fixed leg; Carlos instead describe step by step relations between the largest
area and the amounts of the smaller areas. Alexander noticed the invariance of the
fixed leg.
To build equilateral triangles on the sides of the triangle ABC, students express
their stay looking Pythagorean relationship by varying the lengths of leg AC and
CB hypotenuse. It is noteworthy response alexander:

6. When pentagons on the sides of the right triangle ABC are constructed striking is
the answer given by Carlos. Stating that the sums of the areas of children
pentagons is the result. Here is associating this sum with the largest area of the
pentagon. This becomes the result.
quickly deduce the Pythagorean relationship to build hexagons and semicircles on
the sides of the triangle ABC varying lengths of a leg or hypotenuse
When semicircles on the sides of the triangle ABC are built quickly deduce the
essential characteristics of the Pythagorean relationship.
Taking advantage of the implementation of the representations of the calculator.
They showed the permanence of the Pythagorean relationship using the table of
values generated from construction. Speed is significant as realize this
relationship. Mayra presents a general appreciation: but Alexander shows step by
step the relationship by observing equality numbers representing areas in each row
of the table.
Subsequently, the graph obtained from presents data on the table. When asked
about what kind of relationship it is peculiar variables have the answer for them.
Who they remain at a perceptual level. As you can be seen in the response given
by Carlos are related in the form of a line.
In trying to infer or predict the algebraic relationship between the variables involved
they could not do so explicitly have an intuitive idea, but the transition to algebraic
representation was not given. the algebraic expression is deducted and interpret it
and persist tracks teacher to write the answers show difficulties in interpreting the
value of it. At least observed features of wanting to generalize the answer.
By exploring the relationship of the length of the hypotenuse and the length
variable hick could look that students analyzed different values using tables and
graphs which evinced a degree of algorithmic flow and step generate new
challenges because the calculator as a tool to enable connections with
mathematical concepts looked.

7. Conclusions
Although the requested level of generalization was not very rigorous students were
able to identify relationships between areas built on the legs and the hypotenuse in
a language consistent with the requirements. Involved some concepts and
invariant properties underlying the Pythagorean Theorem.
The ability to dynamically explore the Pythagorean relationship is often presented
from the geometric algebra allowed to look at the tabular representation and
animation of geometric objects genre motivation and interest to further deepen its
study observing it from the geographical point of view.
Mediated experience technological tool, could reveal the difficulties and conceptual
gaps that have the most students in the identification and characterization of
geometric objects (confusion between segment - line, dot-segment, long-area).
Bibliographic references
M.EN. math curriculum guidelines. Bogota. Publishing cooperative teaching, 1990.
Memoirs. National teacher training seminar: use of new technologies in the
mathematics classroom. Bogotá, 2002.
Project incorporating new technologies into the curriculum of mathematics
education of Colombia. Phase expansion and deepening. Direction average quality
of preschool education, basic and. 2001
Orozco. Juan Carlos. Modules: systematization of educational experiences.
Document M.E.N., 2003 study.
Recaman Santos, Bernardo. Numbers, a story to tell. Bogota: Taurus, 2002.