## Teorema de Tales:

Si dos líneas coplanarias (que están en el mismo plano) son cortadas por un haz de rectas paralelas entre sí y también coplanarias, los segmentos determinados en ambas rectas son proporcionales. Usamos el teorema de Tales para dividir segmentos en partes iguales o proporcionales.

## Teorema de la altura:

En Geometría, decimos que dos figuras son “semejantes” cuando sus ángulos son iguales y, por lo tanto, sus lados son proporcionales entre sí.
El teorema de la altura describe la relación entre la altura sobre la hipotenusa y los dos segmentos en que ésta queda dividida por dicha altura. En un triángulo rectángulo la altura sobre la hipotenusa es media proporcional entre los dos segmentos en que la hipotenusa queda dividida por dicha altura. ## Proporción áurea:

Aquí podéis ver un documental muy interesante acerca de la importancia de la proporción áurea:

De forma simple, la Proporción Aurea establece que lo pequeño es a lo grande como lo grande es al todo. Habitualmente esto se aplica a las proporciones entre segmentos.  Dos cantidades están en Proporción Áurea si su relación es la misma que la relación de su suma a la mayor de las dos cantidades. El número de oro es un número irracional que se representa mediante la letra griega en honor del escultor griego Phidias.

Su expresión algebráica, para cantidades a y b, siendo a > b Golden ratio

Donde la letra griega phi  ( ) representa la proporción áurea. Su valor es: En éste tema vamos a aprender a obtenerla gráficamente: ## Para saber más….

Tales de Mileto Thales de Mileto (630 a.c al 545 a.c.) (También se puede escribir Tales de Mileto) fue un matemático griego considerado también como el primer filósofo occidental.

Fue el primero de los Siete Sabios de Grecia y uno de los grandes matemáticos de su época. Una de estas aportaciones a la geometría es el llamado “Teorema de Thales“.

.

# UNIT 3: FUNDAMENTAL CONSTRUCTIONS

## EXERCISE 01: SEGMENTS

In geometry, a line segment is a part of a line that is bounded by two distinct endpoints, and contains every point on the line between its end points. Line segments are generally labeled with two capital letters corresponding to their endpoints.

The addition of two segments is another segment that begins at the origin of the first segment and ends at the end of the second segment.
We use this exercise if we have two segments and we want to draw a segment whose length is the addition of the measures of those two segments. STEPS:

1. Draw a line (r).
2. Draw a point A on it.
3. Measure the given segment AB with your compass.
4. Draw an arc from A with that measure, so you get B.
5. Measure the given segment CD with your compass.
6. Draw an arc from B with that measure, so you get D.
7. The solution is the segment AD.

## 2. SUBTRACTION SEGMENTS

We use this exercise if we have two given segments and we want to draw a segment whose measure is the substraction of the measures of those two segments. STEPS:

1. Draw a line (r).
2. Measure the longest segment with your compass, in our case is the segment CD.
3. Draw a point C on r.
4. Draw an arc from dot C with the previous measure (the segment CD), so you get D.
5. Measure the smallest segment with your compass, in our case is the segment AB.
6. Draw an arc from D with that measure, so you get B.
7. The solution is the segment CB.

## 3. DIVIDE A SEGMENT IN PROPORTIONAL PARTS TO THE GIVEN SEGMENTS. STEPS:

1. Draw a segment and call it AB.
2. Draw an oblique ray (r) to the segment from A.
3. Now we are going to divide segment AB into proportional parts to the given segments CD, DE and EF. We know the measures of these segments.
4. Take the measure of the given segment CD with your compass.
5. Draw an arc from dot A with this measure and where this arc crosses the oblique ray we get a point.
6. Take the measure of the given segment DE with your compass.
7. Draw an arc from the point and where this arc crosses the ray we get another point.
8. Take the measure of the given segment EF with your compass.
9. Draw an arc from last point and where this arc crosses the ray we get another point.
10. Now we need to join this last point with dot B.
11. Using our set square draw parallel lines to that segment from the other points on the ray.
12. Where these lines cross the segment AB we get the new segments C’D’, D’E’ and E’F’.

According to Thales’s theorem the segment C’D’ is proportional to segment CD and it will be the same with segment DE and D’E’ and EF with E’F’.

## 4. DIVIDE A SEGMENT IN EQUAL PARTS

Using Thales we can divide a segment in equal parts. STEPS:

1. Draw the given segment AB. This is the segment that we want to divide.
2. From point A draw an oblique ray (r).
3. Chose a measure with your compass and from point A draw arcs on the oblique ray as many arcs as parts you need.
4. Join the last point of the oblique ray with point B.
5. Draw parallels using your set square to the segment B7 from the other points on the ray.

Here we have divided the segment in seven parts, but you can divide the segment in as many parts as you need.

# UNIT 03: FUNDAMENTAL CONSTRUCTIONS

## Thales’ theorem:

If two lines which intersect each other and are in the same plane are cut by parallel lines, segments determined in one line are proportional to the segments determined in the other line. We use the Thales’ theorem to divide segments into equal or proportional parts.

Thales of Miletus There is considerable agreement that Thales was born in Miletus in Greek Ionia in the mid 620s BCE and died in about 546 BCE, but even those dates are indefinite. Greek philosopher who is considered the founder of Greek science, mathematics, and philosophy. He visited Egypt and probably Babylon, bringing back knowledge of astronomy and geometry. He invented deductive mathematics. To him is attributed Thales’ theorem. It is also attributed to Thales the prediction of a Solar Eclipse and more theorems.

# UNIT 2: FUNDAMENTAL CONSTRUCTIONS

## 1. PARALLEL LINES EQUIDISTANT TO A GIVEN LINE STEPS:
First of all we need to draw a line and call it r. We also need to draw a segment and call it AB, this will be the distance between the parallel lines.

1. Take any two points from the given line (r) and call them M and M.
2. Draw two perpendicular lines to the given line (r) passing through points M and M.
3. With our compass we take the measure of segment AB and we translate that distance to the perpendicular lines obtaining points A, B, C and D.
4. Join points A and B and you will get s, one of the parallel lines to the given line r.
5. Join points C and D and you will get t, the other parallel line to the given line r.

## 2. DIVIDE A SEGMENT IN PARTS PROPORTIONAL TO GIVEN ONES USING THALES THEOREM STEPS:
First of all, we need to draw a segment and call it AB. We also need to draw three different segments and we will call them CD, DE and EF.

1. Draw an oblique ray (r) to the segment from its endpoint A.
2. Now we are going to divide segment AB into proportional parts to the given segments CD, DE and EF. We know the measures of those segments.

3. Take the measure of the given segment CD with your pair of compasses and translate it to the ray r from point A.
4. Take the measure of the given segment DE with your pair of compasses and translate it to the ray r from the endpoint of segment CD.
5. Take the measure of the given segment EF with your pair of compasses and translate it to the ray r from the endpoint of segment DE.
6. Now we need to join the endpoint of the segment EF to point B. By doing this, we get a new segment FB.
7. Using our set square draw parallel lines to segment FB from the other points (E and D).
8. Where these lines cross the segment AB we get the new segments C’D’, D’E’ and E’F’.

According to Thales’s theorem the segment C’D’ is proportional to segment CD and it will be the same with segments DE and D’E’ and EF with E’F’.

## 3. DIVIDE A SEGMENT IN EQUAL PARTS USING THALES THEOREM STEPS:
First of all, we need to draw a segment and call it AB.

1. From point A draw an oblique ray (r).
2. Choose a measure with your compass and from point A translate that measure as many times as you want depending on the number parts you want to divide the segment into.
3. Join the last point, in our case 7, with point B.
4. Draw parallels to the segment B7 using your set square from the other points (1, 2, 3, 4, 5 and 6).
5. Where those parallel lines cross segment AB we will get the points which divide segment AB in the desired parts.

## 4. RIGHT TRIANGLE ALTITUDE THEOREM STEPS:
First of all you need to draw the segment AB whose measure is the addition of segments AP and PB.

1. Get the midpoint (O) of segment AB, to get this point you need to draw the line bisector of segment AB.
2. Center your compass in point O and draw an arc whose radius is segment OA or OB.
3. Draw the perpendicular line to segment AB passing through point P.
4. Where that line intersects the arc you have drawn you will get point C.

The measure of the segment PC (x) is the geometric mean of segments a and b.
= a * b

## 5. GOLDEN RATIO

If we want to get the golden ratio of a given segment: STEPS:
First of all we need to draw the given segment, in our case 6 cm.

1. Get the midpoint (O) of segment AB, to get this point you need to draw the line bisector of segment AB.
2. Draw the perpendicular line to segment AB passing through point B.
3. Draw an arc centering your compass in point B whose radius is segment OB.
4. That arc intersects the perpendicular line in point C.
5. Join point C with point A.
6. Center your compass in point C and draw an arc whose measure is segment CB and draw an arc.
7. That arc intersects segment AC in point D.
9. Where that arc intersects the original segment we will get the division of that segment in two that are in golden ratio: a and b.

## 6. GOLDEN RECTANGLE

If we want to draw a rectangle whose sides are in golden ratio: STEPS:
To begin with, we will chose any measure, u, in our case 2 cm., that we will use to get our rectangle.

1. We need to draw a segment whose measure is 2u.
2. At the endpoint of that segment we draw a perpendicular line to it.
3. Centering our compass at that endpoint we draw an arc whose radius is 2u.
4. Where that arc intersects the perpendicular line we get a point.
5. Center your compass in the midpoint of the segment 2u and open it to the point that we have got previously.
6. With that measure draw an arc that intersects the extension of the segment 2u in a point, so we will get the measure of the long side of our rectangle.

You can draw now an square inside the auxiliar rectangle next the original square, so you will get new rentangles in golden ratio where you will be able to draw the golden spiral: Here you can see the solution of the complete drawing sheet: # UNIT 02: FUNDAMENTAL CONSTRUCTIONS

## Thales’ theorem:

If two lines which intersect each other and are in the same plane are cut by parallel lines, segments determined in one line are proportional to the segments determined in the other line. We use the Thales theorem to divide segments into equal or proportional parts.

## Right triangle altitude theorem:

In Geometry, we say that two or more figures are “similar” when all of their angles are equal so their sides are proportional.
The right triangle altitude theorem or geometric mean theorem is a result in elementary geometry that describes a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states that in a right triangle, the altitude on the hypotenuse is the geometric mean to the two segments into which the hypotenuse is divided. ## Golden ratio:

In mathematics and the arts, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to their maximum. The golden ratio is also called the golden section (Latin: sectio aurea) or golden mean. The golden number is an irrational number and it is represented by the Greek letter in honor to Greek sculptor Phidias.

Expressed algebraically, for quantities a and b with a > b Golden ratio

Where the Greek letter  phi ( ) represents the golden ratio. Its value is: In this unit you will learn how to get this ratio graphically: Thales of Miletus There is considerable agreement that Thales was born in Miletus in Greek Ionia in the mid 620s BCE and died in about 546 BCE, but even those dates are indefinite. Greek philosopher who is considered the founder of Greek science, mathematics, and philosophy. He visited Egypt and probably Babylon, bringing back knowledge of astronomy and geometry. He invented deductive mathematics. To him is attributed Thales’ theorem. It is also attributed to Thales the prediction of a Solar Eclipse and more theorems.

# SEGMENTS

In geometry, a line segment is a part of a line that is bounded by two distinct endpoints, and contains every point on the line between its end points. Line segments are generally labeled with two capital letters corresponding to their endpoints.

The addition of two segments is another segment that begins at the origin of the first segment and ends at the end of the second segment.
We use this exercise if we have two segments and we want to draw a segment whose length is the addition of the measures of those two segments. STEPS:

1. Draw a line (r).
2. Draw a point A on it.
3. Measure the given segment AB with your compass.
4. Draw an arc from A with that measure, so you get B.
5. Measure the given segment CD with your compass.
6. Draw an arc from B with that measure, so you get D.
7. The solution is the segment AD.

## 2. SUBTRACTION SEGMENTS

We use this exercise if we have two given segments and we want to draw a segment whose measure is the substraction of the measures of those two segments. STEPS:

1. Draw a line (r).
2. Measure the longest segment with your compass, in our case is the segment CD.
3. Draw a point C on r.
4. Draw an arc from dot C with the previous measure (the segment CD), so you get D.
5. Measure the smallest segment with your compass, in our case is the segment AB.
6. Draw an arc from D with that measure, so you get B.
7. The solution is the segment CB.

## 3. THALES THEOREM

If two lines which are in the same plane are cut by parallel lines, segments determined in one line are proportional to the segments determined in the other line. We use the Thales theorem to divide segments into equal or proportional parts.

## 4. DIVIDE A SEGMENT IN PROPORTIONAL PARTS TO THE GIVEN SEGMENTS. STEPS:

1. Draw a segment and call it AB.
2. Draw an oblique ray (r) to the segment from A.
3. Now we are going to divide segment AB into proportional parts to the given segments CD, DE and EF. We know the measures of these segments.
4. Take the measure of the given segment CD with your compass.
5. Draw an arc from dot A with this measure and where this arc crosses the oblique ray we get a point.
6. Take the measure of the given segment DE with your compass.
7. Draw an arc from the point and where this arc crosses the ray we get another point.
8. Take the measure of the given segment EF with your compass.
9. Draw an arc from last point and where this arc crosses the ray we get another point.
10. Now we need to join this last point with dot B.
11. Using our set square draw parallel lines to that segment from the other points on the ray.
12. Where these lines cross the segment AB we get the new segments C’D’, D’E’ and E’F’.

According to Thales’s theorem the segment C’D’ is proportional to segment CD and it will be the same with segment DE and D’E’ and EF with E’F’.

## 5. DIVIDE A SEGMENT IN EQUAL PARTS

Using Thales we can divide a segment in equal parts. STEPS:

1. Draw the given segment AB. This is the segment that we want to divide.
2. From point A draw an oblique ray (r).
3. Chose a measure with your compass and from point A draw arcs on the oblique ray as many arcs as parts you need.
4. Join the last point of the oblique ray with point B.
5. Draw parallels using your set square to the segment B7 from the other points on the ray.

Here we have divided the segment in seven parts, but you can divide the segment in as many parts as you need.