UNIT 2: BASIC ELEMENTS OF GEOMETRY

1. LINE BISECTOR

The line bisector is a perpendicular line that passes through the midpoint of the segment, so it divides the segment in two equal parts.

STEPS:
First of all we need to draw a segment. We call it AB.

1. Center your compass in point A, open it further from the middle of the segment AB, and draw an arc.
2. Do the same from point B, where these arcs cross each other we get points 1 and 2.
3. Join 1 and 2, and this way we will get the line bisector of segment AB.

2. PERPENDICULAR LINE TO A LINE FROM A POINT ON IT

Given a line and a point on that line, we will construct a perpendicular line through the given point.
We say that a line is perpendicular to other line when they intersect forming a right angle (90º).

STEPS:
First of all we need to draw a line (r) and mark a point (A) on it.

1. Center your compass in the given point A and draw an arc with the measure you want, where the arc crosses the line we get 1 and 2
2. Get the line bisector between 1 and 2
3. Join 3 and 4, and this way we will get the perpendicular to the given line on point A.

3. PERPENDICULAR LINE TO A LINE FROM AN EXTERNAL POINT

Given a line and a point outside that line, we will construct a perpendicular line through the given point.
We say that a line is perpendicular to other line when they intersect forming a right angle (90º).

STEPS:
First of all we need to draw a line (r) and mark an external point (A). It doesn’t matter where the point is, below or above the line, the steps will be the same.

1. Center your compass in the given point A and draw an arc which crosses the given line r two points called 1 and 2.
2. Get the line bisector between 1 and 2.
3. Join 3 and 4, and this way we will get the perpendicular to the given line on point A.

4. PERPENDICULAR LINE TO A GIVEN RAY ON ITS ENDPOINT

We say that a line is perpendicular to other line when they intersect forming a right angle (90º).

STEPS:
First of all we need to draw a ray (r) and call its endpoint A.

1. Center your compass in the endpoint of the ray (A). Draw an arc with the measure that you want and where this arc crosses the ray we get point 1.
2. Center your compass in point 1 and with the previous measure draw another arc. Where that arc crosses the previous one we get point 2.
3. Center your compass in point 2 and with the same measure draw another arc. Where that arc crosses the first arc we have drawn, we get point 3.
4. Center your compass in point 3 and with the same measure draw another arc. Where that arc crosses the last arc you have drawn, we get point 4.
5. Joining point 4 with the given point A we will get the perpendicular line to the ray on its endpoint.

MANEJO DE LA ESCUADRA Y EL CARTABÓN

Hay que manejar la escuadra y cartabón con soltura y suavidad, sin ejercer sobre ellas una excesiva presión, pero sí la necesaria para evitar todo movimiento.
Para trazar paralelas a una dirección debemos proceder de la siguiente manera:

1. Se coloca la hipotenusa (lado más largo) de la escuadra coincidiendo con la recta a la que queremos trazar paralelas.
2. Se apoya en un cateto de la escuadra la hipotenusa del cartabón.
3. Se fija el cartabón y se desplaza la escuadra, trazando por su hipotenusa las paralelas deseadas.

Si queremos trazar perpendiculares a una dirección tendremos que:

1. Se coloca la hipotenusa (lado más largo) de la escuadra coincidiendo con la recta a la que queremos trazar paralelas.
2. Se apoya en un cateto de la escuadra la hipotenusa del cartabón.
3. Fijando el cartabón, se gira la escuadra sin levantarla del papel hasta que apoyemos el otro cateto sobre el cartabón.
4. Trazar la perpendicular por la hipotenusa de la escuadra.

HANDLING THE SET SQUARE

How should we handle the set square?

You have to handle your set square softly and with accuracy without exercising too much pressure on them, only the needed one to avoid movement.
To draw parallel lines to one direction we have to follow these steps:

1. The 45 set square hypotenuse (longest side) is placed attached to the line to which we want to draw the parallels.
2. The 60-30 set square hypotenuse is attached to the 45 set square leg.
3. Fix the 60-30 set square and move the 45 set square upwards or downwards drawing the desired parallel lines along its hypotenuse.

If we want to draw perpendicular lines to one direction, we will have to follow the first two steps as stated for parallel lines and then the following ones:

1. Having fixed the 60-30 set square, the 45 set square is turned until the other leg is attached to the hypotenuse of the 60-30 set square.
2. Draw the perpendicular line along the hypotenuse of the 45 set square.

UNIT 01: BASIC ELEMENTS OF PLASTIC EXPRESSION

EXERCISE 02: PARALLEL LINES

How do we use the set square?

You have to handle your set square softly and with accuracy without exercising too much pressure on them, only the needed one to avoid movement.
To draw parallel lines to one direction we have to follow these steps:

1. The 45 set square hypotenuse (longest side) is placed attached to the line to which we want to draw the parallels.
2. The 60-30 set square hypotenuse is attached to the 45 set square leg.
3. Fix the 60-30 set square and move the 45 set square upwards or downwards drawing the desired parallel lines along its hypotenuse.

If we want to draw perpendicular lines to one direction, we will have to follow the first two steps as stated for parallel lines and then the following ones:

1. Having fixed the 60-30 set square, the 45 set square is turned until the other leg is attached to the hypotenuse of the 60-30 set square.
2. Draw the perpendicular line along the hypotenuse of the 45 set square.

UNIT 1: GEOMETRIC CONSTRUCTIONS

1. LINE BISECTOR

It is the locus of the points in the plane equidistant from the endpoints of a segment. Therefore it is the locus of all the circumference centres that passes through these endpoints.

The line bisector is a perpendicular line that passes through the midpoint of the segment.

STEPS:
First of all we need to draw a segment. We call it AB.

1. Center your compass in point A, open it further from the middle of the segment AB, and draw an arc.
2. Do the same from point B, where these arcs cross each other we get points 1 and 2.
3. Join 1 and 2, and this way we will get the line bisector of segment AB.

2. PERPENDICULAR LINE TO A LINE FROM A POINT ON IT

Given a line and a point on that line, we will construct a perpendicular line through the given point.
We say that a line is perpendicular to other line when they intersect forming a right angle.

STEPS:
First of all we need to draw a line (r) and mark a point (A) on it.

1. Center your compass in the given point A and draw an arc with the measure you want, where the arc crosses the line we get 1 and 2
2. Get the line bisector between 1 and 2
3. Join 3 and 4, and this way we will get the perpendicular to the given line on point A.

3. PERPENDICULAR LINE TO A LINE FROM AN EXTERNAL POINT

Given a line and a point outside that line, we will construct a perpendicular line through the given point.
We say that a line is perpendicular to other line when they intersect forming a right angle (90º).

STEPS:
First of all we need to draw a line (r) and mark an external point (A). It doesn’t matter where the point is, below or above the line, the steps will be the same.

1. Center your compass in the given point A and draw an arc which crosses the given line r two points called 1 and 2.
2. Get the line bisector between 1 and 2.
3. Join 3 and 4, and this way we will get the perpendicular to the given line on point A.

4. PERPENDICULAR LINE TO A GIVEN RAY ON ITS ENDPOINT

We say that a line is perpendicular to other line when they intersect forming a right angle (90º).

STEPS:
First of all we need to draw a ray (r) and call its endpoint A.

1. Center your compass in the endpoint of the ray (A). Draw an arc with the measure that you want and where this arc crosses the ray we get point 1.
2. Center your compass in point 1 and with the previous measure draw another arc. Where that arc crosses the previous one we get point 2.
3. Center your compass in point 2 and with the same measure draw another arc. Where that arc crosses the first arc we have drawn, we get point 3.
4. Center your compass in point 3 and with the same measure draw another arc. Where that arc crosses the last arc you have drawn, we get point 4.
5. Joining point 4 with the given point A we will get the perpendicular line to the ray on its endpoint.

5. PARALLEL LINE TO A LINE FROM AN EXTERNAL POINT I

We say that a line is parallel to another line when these two lines never cross each other.

STEPS:
First of all we draw a line (r) and draw an external point to it (A)

1. Center your compass in any point of the line (O) and draw an arc that passes through point A.
2. This arc will cross the given line (r) in two points; we will call them P and Q.
3. Draw an arc which radio is the distance between points Q and A taking P as the center. Where that arc crosses the previous one we will get point B.
4. Join point B with the given point A and you will get p, the parallel line to the given line r.

6. PARALLEL LINE TO A LINE FROM AN EXTERNAL POINT II

We say that a line is parallel to another line when these two lines never cross each other.

STEPS:
First of all we draw a line (r) and draw an external point to it (A)

1. Take any two points from the given line (r) and call them P and Q.
2. Draw a circle which radio is the distance between points P and Q taking A as the center.
3. Draw an arc which radio is the distance between points P and A taking Q as the center. Where this arc crosses the circle we will get point B.
4. Join point B with the given point A and you will get p, the parallel line to the given line r.

TEMA 1: CONSTRUCCIONES GEOMÉTRICAS

1. MEDIATRIZ DE UN SEGMENTO

Es el lugar geométrico de los puntos del plano que equidistan de los extremos de un segmento. Por lo tanto es el lugar geométrico de todos los centros de las circunferencias que pasan por dichos extremos.

PASOS:

1. Hacemos centro de compás en A con radio mayor que la mitad de AB.
2. Trazamos dos arcos de circunferencia.
3. Hacemos centro de compás en B con el mismo radio.
4. Trazamos dos arcos de circunferencia.
5. Donde corten a los trazados previamente obtenemos los puntos 1 y 2.
6. Uniendo 1 y 2 obtenemos la mediatriz del segmento AB.

2. RECTA PERPENDICULAR A OTRA POR UN PUNTO SITUADO EN ELLA

Una recta es perpendicular a otra cuando forma con ella un ángulo recto (90º)

PASOS:

1. Teniendo la recta (r) y un punto situado en ella (A)
2. Trazamos un arco de circunferencia tomando como centro A.
3. Ese arco de circunferencia cortará a la recta (r) en dos puntos 1 y 2.
4. Trazamos la mediatriz del segmento marcado en la recta (r) 12.
5. Uniendo los puntos 3 y 4 obtenemos la recta perpendicular a (r) que pasa por el punto A.

3. RECTA PERPENDICULAR A OTRA POR UN PUNTO EXTERIOR

Una recta es perpendicular a otra cuando forma con ella un ángulo recto (90º)

PASOS:

1. Teniendo la recta (r) y un punto exterior a ella (A)
2. Trazamos un arco de circunferencia tomando como centro A.
3. Ese arco de circunferencia cortará a la recta (r) en dos puntos 1 y 2.
4. Trazamos la mediatriz del segmento marcado en la recta (r) 12.
5. Uniendo los puntos 3 y 4 obtenemos la recta perpendicular a (r) que pasa por el punto A.

4. RECTA PERPENDICULAR A OTRA POR UNO DE SUS EXTREMOS

Una recta es perpendicular a otra cuando forma con ella un ángulo recto (90º)

PASOS:

1. Hacemos centro de compás en el punto dado A.
2. Trazamos un arco de circunferencia y donde corte a la recta obtenemos 1
3. Haciendo centro de compás en 1 con la misma abertura obtenemos 2.
4. Haciendo centro de compás en 2 con la misma abertura obtenemos 3.
5. Hacemos centro de compás en 2 y en 3 con la misma abertura y obtenemos 4.
6. Unimos 4 con A y obtenemos la recta perpendicular a la dada que pasa por uno de sus extremos.

5. RECTA PARALELA A OTRA POR UN PUNTO EXTERIOR – MÉTODO I

Una recta es paralela a otra cuando nunca se corta con esta.

PASOS:

1. Tomando centro de compás en un punto cualquiera de la recta dada (r) al que llamamos O trazamos un arco de circunferencia que pase por el punto dado A
2. Ese arco de circunferencia cortará a la recta (r) en dos puntos P y Q
3. Hacemos arco de circunferencia en P, tomando como radio la distancia QA
4. Donde corte al primer arco de circunferencia trazado obtenemos B
5. Uniendo B con A obtenemos la recta paralela a r que pasa por A.

6. RECTA PARALELA A OTRA POR UN PUNTO EXTERIOR – MÉTODO II

Una recta es paralela a otra cuando nunca se corta con esta.

PASOS:

1. Tomamos dos puntos aleatorios de la recta dada (r) y los llamamos P y Q
2. Hacemos centro en A con radio de circunferencia PQ y dibujamos la circunferencia.
3. Hacemos centro en Q con radio de circunferencia PA y donde corte a la primera circunferencia obtenemos B
4. Uniendo B con A obtenemos la recta paralela a la recta dada que pasa por A