PROPORCIONALIDAD

TEMA 02: PROPORCIONALIDAD

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.
THALES-THEOREM
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.
SIMILAR-TRIANGLES

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  \varphi en honor del escultor griego Phidias.

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

Golden ratio

Donde la letra griega phi  (\varphi) 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“.

.

EXERCISE 02: CENTERS OF TRIANGLES

UNIT 6: TRIANGLES

EXERCISE 02: CENTERS OF TRIANGLES

1. CIRCUMCENTER

The circumcenter is the center of the triangle’s circumcircle, one that passes through the three vertices of the triangle.
It is located at the intersection of the three side bisectors. As the three sides of a triangle are segments, if we draw the three side bisectors, we will get a point that will be the center of a circle that passes through the three vertices of the triangle, the Circumcircle.
It is called graphically with the letter O.

  • In an acute triangle, the circumcircle’s center is inside the triangle.
  • In an obtuse triangle, the circumcircle’s center is outside the triangle.
  • In a right triangle, the circumcircle’s center is the midpoint of the hypotenuse.

CIRCUMCENTER
STEPS:

  1. Draw a triangle and call its vertices ABC.
  2. Draw the line bisector of side AB.
  3. Draw the line bisector of side BC.
  4. Draw the line bisector of side AC.
  5. The three line bisectors intersect at a point, which we call O, the circumcenter.
  6. If we make a circle centering the compass at the circumcenter and opening it to one of the vertices of the triangle, this circle will necessarily pass by the other two. This is the circumcircle of the triangle.

2. INCENTER

The incenter is the center of the triangle’s incircle, also known as inscribed circle, it is the largest circle that will fit inside the triangle. Each of the triangle’s three sides is a tangent to the circumference.
It is located at the intersection of the three angle bisectors. As a triangle has three angles, if we draw the three angle bisectors, we will get a point that will be the center of a circle that is tangent to the three sides of the triangle, the Incircle.
It is called graphically with the letter I.
Incenter is always inside the triangle.
INCENTER
STEPS:

    1. Draw a triangle and call its vertices ABC.
    2. Draw the angle bisector of angle A.
    3. Draw the angle bisector of angle B.
    4. Draw the angle bisector of angle C.
    5. The three angle bisectors intersect at a point, which we call I, the incenter.

To know the radious of the incircle and be able to draw it we need to know the least distance from the incenter to the sides of the triangle.
The least distance from a point to a line is the perpendicular.
If we draw the perpendicular line from the incenter to any of the sides of the triangle, we will get the radious of the incircle.

    1. Draw the perpendicular line, for example, to side AC passing through point I.

Where this line crosses side AC, we get T, the tangency point to this side.

  • If we make a circle centering the compass at the incenter and opening it to point T, this circle will necessarily be tangent to the other two sides. This is the incircle of the triangle.

 

3. CENTROID

The barycenter or centroid is located on the intersection of the three medians of the triangle, and is equivalent to the center of gravity of the triangle.
It is called graphically with the letter G.
The medians of the triangle are each segments which join each vertex of the triangle with the midpoint of the opposite side.
Barycenter is always inside the triangle.
BARYCENTER
STEPS:

    1. Draw a triangle and call its vertices ABC.
    2. Draw the perpendicular bisector of side AC, so you get the midpoint of this side MAC.
    3. Join MAC to the oposite vertex B, this is the median of vertex B.
    4. Draw the perpendicular bisector of side AB, so you get the midpoint of this side MAB.
    5. Join MAB to the oposite vertex C, this is the median of vertex C.
    6. Draw the perpendicular bisector of side BC, so you get the midpoint of this side MBC.
    7. Join MBC to the oposite vertex A, this is the median of vertex A.

The three medians intersect at a point, which we call G, the barycenter.

4. ORTHOCENTER

The Orthocenter is the point located at the intersection of the triangle heights.
It is called the height of a triangle the segment which join a vertex of a triangle to the opposite side -or its extension- forming a right angle (90 °). All triangles have three heights.
It is called graphically with the letter H.

  • In an acute triangle, the orthocenter is inside the triangle.
  • In an obtuse triangle, the orthocenter is outside the triangle.
  • In a right triangle, the orthocenter is the vertex of the triangle’s right angle.

ORTHOCENTER
STEPS:

  1. Draw a triangle and call its vertices ABC.
  2. Draw the perpendicular line to side AC passing through point B. This is the B’s height.
  3. Draw the perpendicular line to side AB passing through point C. This is the C’s height.
  4. Draw the perpendicular line to side BC passing through point A. This is the A’s height.
  5. The three heights intersect at a point, which we call H, the orthocenter.

Here you can see the solution of the complete drawing sheet:

CENTERS-TRIANGLE

THE CENTERS OF TRIANGLES

UNIT 06: TRIANGLES

THE CENTERS OF TRIANGLES

Journey to the Center of a Triangle

Here you can see a film, ‘Journey to the Center of a Triangle’ (1976), created by Bruce & Katharine Cornwell to help you understanding basic ideas about the centers of triangles.
The film created on the Tektronics 4051 Graphics Terminal. Presents a series of animated constructions that determine the center of a variety of triangles, including such centers as circumcenter, incenter, centroid and orthocenter. This movie is part of the collection: Academic Film Archive of North America.


Points of concurrency: The point where three or more lines intersect.
POINTS-CONCURRENCY
In the figure above the three lines all intersect at the same point P – called the point of concurrency.

1. CIRCUMCENTER

The circumcenter is the center of the triangle’s circumcircle, one that passes through the three vertices of the triangle.
Located at the intersection of the three side bisectors.
It is called graphically with the letter O.

  • In an acute triangle, the circumcircle’s center is inside the triangle.
  • In an obtuse triangle, the circumcircle’s center is outside the triangle.
  • In a right triangle, the circumcircle’s center is the midpoint of the hypotenuse.

CIRCUMCENTER

2. INCENTER

The incenter is the center of the triangle’s incircle, also known as inscribed circle, it is the largest circumference that will fit inside the triangle. Each of the triangle’s three sides is a tangent to the circumference.
Located at the intersection of the three angle bisectors.
It is called graphically with the letter I.
Incenter is always inside the triangle.
INCENTER

3. BARYCENTER

The barycenter or centroid is located on the intersection of the three medians of the triangle, and is equivalent to the center of gravity of the triangle.
The medians of the triangle are each segments which join each vertex of the triangle with the midpoint of the opposite side.
It is called graphically with the letter G.
Barycenter is always inside the triangle.
BARYCENTER

4. ORTHOCENTER

The Orthocenter is the point located at the intersection of the triangle heights.
It is called the height of a triangle the segment which join a vertex of a triangle to the opposite side -or its extension- forming a right angle (90 °). All triangles have three heights.
It is called graphically with the letter H.

  • In an acute triangle, the orthocenter is inside the triangle.
  • In an obtuse triangle, the orthocenter is outside the triangle.
  • In a right triangle, the orthocenter is the vertex of the triangle’s right angle.

ORTHOCENTER

 

 

TRIANGLES

UNIT 6: TRIANGLES

EXERCISE 01: TRIANGLES

1. EQUILATERAL TRIANGLE (l = 5 cm.)


STEPS:

  1. Draw a side.
  2. Center your compass in point A and open it to point B, draw an arc.
  3. Now center the compass at the point B and open it to point A, draw an arc.
  4. Mark the arc intersection point C.
  5. Join point A with point C and do the same with point B so you get triangle.

2. ISOSCELES TRIANGLE (a = b = 4,5 cm., c = 3,5 cm.)


STEPS:

  1. Draw the different side.
  2. Measure the equal side with your compass (4,5 cm.).
  3. With this measure draw an arc centering your compass in point A.
  4. With the same measure draw another arc centering your compass in point B.
  5. Mark the arc intersection point C.
  6. Join point A with point C and do the same with point B so you get triangle.

3. SCALENE TRIANGLE (a = 7 cm., b = 4,5 cm., c = 5,5 cm.)


STEPS:

  1. Draw a side, for example side a = 7 cm.
  2. Measure another side with your compass, for example b = 4,5 cm.
  3. With this measure draw an arc centering your compass in point C.
  4. Measure the last side with your compass, c = 5,5 cm.
  5. With this measure draw an arc centering your compass in point B.
  6. Mark the arc intersection point A.
  7. Join point A with point C and do the same with point B so you get triangle.

4. RIGHT TRIANGLE (a = 6 (Hypotenuse), b = 3 cm.)


STEPS:

  1. Draw the side b = 3 cm.
  2. Draw a perpendicular line from point A.
  3. Measure side a with your compass, a = 6 cm.
  4. With this measure draw an arc centering your compass in point C.
  5. Where this arc crosses the perpendicular line to point A we get point B.
  6. Join point B with point C so you get triangle.

Practice triangles:

Here you can download a .pdf and then print in a DIN-A4, to practice triangles exercises.
PRACTICE_TRIANGLES

TRIANGLES

UNIT 06: TRIANGLES

TRIANGLES

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments.

The sum of all the internal angles in the triangle is 180º.

How to label a triangle

  • The vertices of the triangle are named with capital letters (A, B, C)
  • The sides of the triangle are named with the letter of its opposite vertex (a, b, c)
  • The angles of the triangle are named with the letter of the vertex and angle symbol (^).

Types of triangles

Triangles are classified two ways:

For their SIDES:

Equilateral
Three equal sides and angles.

Isosceles
Two equal sides and angles.

Scalene
No equal sides or angles.

For their INTERNAL ANGLES

Acute
All their angles are ACUTE.

Right
One RIGHT ANGLE.

In right triangles, the sides that form the right angle are called LEGS and the side that faces them is called HYPOTENUSE.

Obtuse
One OBTUSE angle.

 

EXERCISE 06: CENTERS OF TRIANGLES

UNIT 05: CENTERS OF TRIANGLES

EXERCISE 06: CENTERS OF TRIANGLES

1. CIRCUMCENTER

The circumcenter is the center of the triangle’s circumcircle, one that passes through the three vertices of the triangle.
It is located at the intersection of the three side bisectors. As we already know the line bisector is the locus of all the circle centers that pass through the endpoints of a segment. As the three sides of a triangle are segments, if we draw the three side bisectors, we will get a point that will be the center of a circle that passes through the three vertices of the triangle, the Circumcircle.
It is called graphically with the letter O.

  • In an acute triangle, the circumcircle’s center is inside the triangle.
  • In an obtuse triangle, the circumcircle’s center is outside the triangle.
  • In a right triangle, the circumcircle’s center is the midpoint of the hypotenuse.

CIRCUMCENTER
STEPS:

  1. Draw a triangle and call its vertices ABC.
  2. Draw the line bisector of side AB.
  3. Draw the line bisector of side BC.
  4. Draw the line bisector of side AC.
  5. The three line bisectors intersect at a point, which we call O, the circumcenter.
  6. If we make a circle centering the compass at the circumcenter and opening it to one of the vertices of the triangle, this circle will necessarily pass by the other two. This is the circumcircle of the triangle.

2. INCENTER

The incenter is the center of the triangle’s incircle, also known as inscribed circle, it is the largest circle that will fit inside the triangle. Each of the triangle’s three sides is a tangent to the circumference.
It is located at the intersection of the three angle bisectors. As we already know the angle bisector is the locus of all the circle centers that are tangent to the sides of an angle. As a triangle has three angles, if we draw the three angle bisectors, we will get a point that will be the center of a circle that is tangent to the three sides of the triangle, the Incircle.
It is called graphically with the letter I.
Incenter is always inside the triangle.
INCENTER
STEPS:

  1. Draw a triangle and call its vertices ABC.
  2. Draw the angle bisector of angle A.
  3. Draw the angle bisector of angle B.
  4. Draw the angle bisector of angle C.
  5. The three angle bisectors intersect at a point, which we call I, the incenter.
  6. To know the radious of the incircle and be able to draw it we need to know the least distance from the incenter to the sides of the triangle.
    The least distance from a point to a line is the perpendicular.
    If we draw the perpendicular line from the incenter to any of the sides of the triangle, we will get the radious of the incircle.

  7. Draw the perpendicular line, for example, to side AC passing through point I.
  8. Where this line crosses side AC, we get T, the tangency point to this side.

  9. If we make a circle centering the compass at the incenter and opening it to point T, this circle will necessarily be tangent to the other two sides. This is the incircle of the triangle.

3. CENTROID

The barycenter or centroid is located on the intersection of the three medians of the triangle, and is equivalent to the center of gravity of the triangle.
It is called graphically with the letter G.
The medians of the triangle are each segments which join each vertex of the triangle with the midpoint of the opposite side.
Barycenter is always inside the triangle.
BARYCENTER
STEPS:

  1. Draw a triangle and call its vertices ABC.
  2. Draw the perpendicular bisector of side AC, so you get the midpoint of this side MAC.
  3. Join MAC to the oposite vertex B, this is the median of vertex B.
  4. Draw the perpendicular bisector of side AB, so you get the midpoint of this side MAB.
  5. Join MAB to the oposite vertex C, this is the median of vertex C.
  6. Draw the perpendicular bisector of side BC, so you get the midpoint of this side MBC.
  7. Join MBC to the oposite vertex A, this is the median of vertex A.
  8. The three medians intersect at a point, which we call G, the barycenter.

4. ORTHOCENTER

The Orthocenter is the point located at the intersection of the triangle heights.
It is called the height of a triangle the segment which join a vertex of a triangle to the opposite side -or its extension- forming a right angle (90 °). All triangles have three heights.
It is called graphically with the letter H.

  • In an acute triangle, the orthocenter is inside the triangle.
  • In an obtuse triangle, the orthocenter is outside the triangle.
  • In a right triangle, the orthocenter is the vertex of the triangle’s right angle.

ORTHOCENTER
STEPS:

  1. Draw a triangle and call its vertices ABC.
  2. Draw the perpendicular line to side AC passing through point B. This is the B’s height.
  3. Draw the perpendicular line to side AB passing through point C. This is the C’s height.
  4. Draw the perpendicular line to side BC passing through point A. This is the A’s height.
  5. The three heights intersect at a point, which we call H, the orthocenter.

Here you can see the solution of the complete drawing sheet:

CENTERS-TRIANGLE

THE CENTERS OF TRIANGLES

UNIT 05: THE CENTERS OF TRIANGLES

Journey to the Center of a Triangle

Here you can see a film, ‘Journey to the Center of a Triangle’ (1976), created by Bruce & Katharine Cornwell to help you understanding basic ideas about the centers of triangles.
The film created on the Tektronics 4051 Graphics Terminal. Presents a series of animated constructions that determine the center of a variety of triangles, including such centers as circumcenter, incenter, centroid and orthocenter. This movie is part of the collection: Academic Film Archive of North America.


Fisrt of all, we need to know what a point of concurrency is, because all the centers of triangles are points of concurrency.

Point of concurrency:

It is the point where three or more lines intersect.
POINTS-CONCURRENCY
In the figure above the three lines all intersect at the same point P – called the point of concurrency.

Centers of triangles

1. CIRCUMCENTER

The circumcenter is the center of the triangle’s circumcircle, one that passes through the three vertices of the triangle.
It is located at the intersection of the three side bisectors. As we already know the line bisector is the locus of all the circle centers that pass through the endpoints of a segment. As the three sides of a triangle are segments, if we draw the three side bisectors, we will get a point that will be the center of a circle that passes through the three vertices of the triangle, the Circumcircle.
It is called graphically with the letter O.

  • In an acute triangle, the circumcircle’s center is inside the triangle.
  • In an obtuse triangle, the circumcircle’s center is outside the triangle.
  • In a right triangle, the circumcircle’s center is the midpoint of the hypotenuse.

CIRCUMCENTER

2. INCENTER

The incenter is the center of the triangle’s incircle, also known as inscribed circle, it is the largest circle that will fit inside the triangle. Each of the triangle’s three sides is a tangent to the circumference.
It is located at the intersection of the three angle bisectors. As we already know the angle bisector is the locus of all the circle centers that are tangent to the sides of an angle. As a triangle has three angles, if we draw the three angle bisectors, we will get a point that will be the center of a circle that is tangent to the three sides of the triangle, the Incircle.
It is called graphically with the letter I.
Incenter is always inside the triangle.
INCENTER

3. BARYCENTER

The barycenter or centroid is located on the intersection of the three medians of the triangle, and is equivalent to the center of gravity of the triangle.
The medians of the triangle are each segments which join each vertex of the triangle with the midpoint of the opposite side.
It is called graphically with the letter G.
Barycenter is always inside the triangle.
BARYCENTER

4. ORTHOCENTER

The Orthocenter is the point located at the intersection of the triangle heights.
It is called the height of a triangle the segment which join a vertex of a triangle to the opposite side -or its extension- forming a right angle (90 °). All triangles have three heights.
It is called graphically with the letter H.

  • In an acute triangle, the orthocenter is inside the triangle.
  • In an obtuse triangle, the orthocenter is outside the triangle.
  • In a right triangle, the orthocenter is the vertex of the triangle’s right angle.

ORTHOCENTER

To learn more…

We have studied four centers of triangles, circumcenter, incenter, centroid and orthocenter. Orthocenter, circumcenter and centroid lay in the same line: the Euler line, named after Leonhard Euler, a Swiss mathematician. Euler line is a line determined from any triangle that is not equilateral. It passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle.

Euler’s line:


Nine-point circle:

nine-points-circle

EXERCISE 05: TRIANGLES II

UNIT 04: TRIANGLES

EXERCISE 05: TRIANGLES II

1. ISOSCELES TRIANGLE

Draw an isosceles Triangle known one of its equal sides b = 4 cm. and one of its equal angles A = 30°.
ISOSCELES

2. ISOSCELES TRIANGLE

Draw an isosceles Triangle known the unequal side c = 5 cm. and the unequal angle C = 45°.
ISOSCELES-02

3. RIGHT TRIANGLE

Draw a right triangle known its legs b = 5 cm. and c = 4 cm.
RIGHT

4. SCALENE TRIANGLE

Draw an scalene triangle known side a = 6 cm., side b = 5 cm. and the angle between them C = 40°.
SCALENE

5. SCALENE TRIANGLE

Draw a triangle known its side a = 5 cm., its angle A = 60° and other of its angles C = 45°.
SCALENE-02

6. SCALENE TRIANGLES

Draw the triangles whose sides measure c = 5 cm. and b = 6 cm. and one of their angles is A = 45°.

SCALENE-03
Here you can download and print the photocopy that I gave you on Monday to continue practising the construction of triangles. Nex week I will share the solutions!
Here you can download the solutions.

EXERCISE 04: TRIANGLES

UNIT 04: TRIANGLES

EXERCISE 04: TRIANGLES

1. EQUILATERAL TRIANGLE (l = 5 cm.)


STEPS:

  1. Draw the given side and call its endpoints A and B.
  2. Center your compass in point A and open it to point B, draw an arc.
  3. Now center the compass in point B and with the same measure, draw another arc.
  4. Where these two arcs intersect each other, we will get point C.
  5. Join point A with point C and do the same with point B so you get the equilateral triangle ABC.

2. ISOSCELES TRIANGLE (a = b = 4.5 cm., c = 3.5 cm.)


STEPS:

  1. Draw the different side, in our case side c whose measure is 3.5 cm.
  2. Take the measure of the other two sides with your compass, in our case 4.5 cm.
  3. With this measure draw an arc centering your compass in point A.
  4. With the same measure draw another arc centering your compass in point B.
  5. Where these two arcs intersect each other, we will get point C.
  6. Join point A with point C and do the same with point B so you get the isosceles triangle ABC.

3. SCALENE TRIANGLE (a = 7 cm., b = 4.5 cm., c = 5.5 cm.)


STEPS:

  1. Draw a side, for example side a = 7 cm.
  2. Measure another side with your compass, for example b = 4.5 cm.
  3. With this measure, draw an arc centering your compass in point C.
  4. Measure the last side with your compass, c = 5.5 cm.
  5. With this measure draw an arc centering your compass in point B.
  6. Where these two arcs intersect each other, we will get point A.
  7. Join point A with point C and do the same with point B so you get the scalene triangle ABC.

4. TRIANGLES (a = 6 cm., b = 4 cm., B = 30º)

In this exercise we can get two possible solutions.

STEPS:

  1. Draw the given side a and call its endpoints B and C.
  2. Draw a 30º angle whose vertex is point B, to do this you have to follow the steps given to draw the trisection of an angle.
  3. Measure side b with your compass, in our case b = 4 cm.
  4. Center your compass in point C and draw an arc. Where this arc interects the side of the 30º angle we will get points A1 and A2.
  5. Bear in mind that in this exercise we can get two possible solutions A1BC and A2 BC.

5. RIGHT TRIANGLE (a = 6 (Hypotenuse), b = 3 cm.)


STEPS:
As we have to draw a right triangle we have to remember that this triangle has a 90º angle and that this angle faces the hypotenuse.

  1. Draw the given side b (3 cm.) and call its endpoints A and C.
  2. Draw a 90º angle whose vertex is point A, to do this you will need to follow the steps given to draw the perpendicular to a ray.
  3. Measure the hypotenuse with your compass, in our case 6 cm.
  4. Center your compass in point C, and where this arc intersects the right angle we will get point B, the last vertex of our triangle.

6. TRIANGLE (a = 6.5 cm., B = 30º, C = 105º)


STEPS:

  1. Draw the given side a and call its endpoints B and C.
  2. Draw a 30º angle whose vertex is point B, to do this you have to follow the steps given to draw the trisection of an angle.
  3. Draw a 105º angle whose vertex is point C, to do this you have to follow the steps given to draw a 105º angle.
  4. Where these two lines intersect you will get vertex A.

Here you can see the solution of the complete drawing sheet:

TRIANGLES

UNIT 04: TRIANGLES

Definition

A triangle ABC is a flat shape limited by three lines which intersect each other two to two, defining the segments a, b and c, which are the sides of the triangle. In order to get those three segments forming a triangle ABC it is necessary that the length of each of those segments is smaller than the addition of the other two and bigger than their subtraction.
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments.
The addition of all the internal angles in the triangle is 180º.

How to label a triangle

  • The vertices of the triangle are named with capital letters (A, B, C)
  • The sides of the triangle are named with the letter of its opposite vertex (a, b, c)
  • The angles of the triangle are named with the letter of the vertex and angle symbol (^).

Types of triangles

Triangles are classified two ways:

Due to their SIDES:

Equilateral
Three equal sides and angles.

Isosceles
Two equal sides and angles.

Scalene
No equal sides or angles.

Due to their INTERNAL ANGLES

Acute
All their angles are ACUTE.

Right
One RIGHT ANGLE.

In right triangles, the sides that form the right angle are called LEGS and the side that faces them is called HYPOTENUSE.

Obtuse
One OBTUSE angle.