# UNIT 1: GEOMETRIC CONSTRUCTIONS

## EXERCISE 01: 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.

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

- 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
- Get the line bisector between 1 and 2
- 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.

- Center your compass in the given point A and draw an arc which crosses the given line r two points called 1 and 2.
- Get the line bisector between 1 and 2.
- 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.

- 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.
- 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.
- 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.
- 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.
- 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)

- Center your compass in any point of the line (O) and draw an arc that passes through point A.
- This arc will cross the given line (r) in two points; we will call them P and Q.
- 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.
- 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)

- Take any two points from the given line (r) and call them P and Q.
- Draw a circle which radio is the distance between points P and Q taking A as the center.
- 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.
- Join point B with the given point A and you will get p, the parallel line to the given line r.

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