# 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.