## EXERCISE 02: 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.

- Take any two points from the given line (r) and call them M and M.
- Draw two perpendicular lines to the given line (r) passing through points M and M.
- 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.
- Join points A and B and you will get s, one of the parallel lines to the given line r.
- 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.

- Draw an oblique ray (r) to the segment from its endpoint A.
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.

- Take the measure of the given segment CD with your pair of compasses and translate it to the ray r from point A.
- 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.
- 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.
- Now we need to join the endpoint of the segment EF to point B. By doing this, we get a new segment FB.
- Using our set square draw parallel lines to segment FB from the other points (E and D).
- 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.

- From point A draw an oblique ray (r).
- 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.
- Join the last point, in our case 7, with point B.
- Draw parallels to the segment B7 using your set square from the other points (1, 2, 3, 4, 5 and 6).
- 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.

- Get the midpoint (O) of segment AB, to get this point you need to draw the line bisector of segment AB.
- Center your compass in point O and draw an arc whose radius is segment OA or OB.
- Draw the perpendicular line to segment AB passing through point P.
- 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.

x² = 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.

- Get the midpoint (O) of segment AB, to get this point you need to draw the line bisector of segment AB.
- Draw the perpendicular line to segment AB passing through point B.
- Draw an arc centering your compass in point B whose radius is segment OB.
- That arc intersects the perpendicular line in point C.
- Join point C with point A.
- Center your compass in point C and draw an arc whose measure is segment CB and draw an arc.
- That arc intersects segment AC in point D.
- Draw an arc centering your compass in point A whose radius is segment AD.
- 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.

- We need to draw a segment whose measure is 2u.
- At the endpoint of that segment we draw a perpendicular line to it.
- Centering our compass at that endpoint we draw an arc whose radius is 2u.
- Where that arc intersects the perpendicular line we get a point.
- Center your compass in the midpoint of the segment 2u and open it to the point that we have got previously.
- 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: