# 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 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 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.