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Basic Geometric Constructions

In the section "Constructions with Straightedge and Compass", we used algebraic techniques to prove that certain geometric constructions using only the tools of the Greek geometers where impossible to form. In that section we mentioned the following basic constructions:

Basic Constructions

1. Bisect a given angle.

2. Construct a line perpendicular to a given line segment L at a given point P on L.

3. Construct a line through a given point P which is parallel to a given line L.

4. Given line segments of lengths l and l', construct a line segment of length ll'.

5. Given a line segment of length l, construct a line segment of length 1/l.

6. Given a line segment of length l, construct a line segment of length square root of L.

Here we will give brief review of the these constructions from basic geometry


Bisect an angle

Directions:

1. Place the point of the compass on the vertex of angle BAC (point A).

2. Stretch the compass to any length so long as it stays ON the angle.

3. Swing an arc with the pencil that crosses both sides of angle BAC. This will create two intersection points with the sides of the angle.

4. Place the point on one of these intersection points created on the sides of the angle BAC. If needed, stretch your compass to a sufficient length to place your pencil well into the interior of the angle. Stay between the sides (rays) of the angle. Place an arc in this interior - you do not need to cross the sides of the angle.

5. Without changing the width of the compass, place the point of the compass on the other intersection point on the side of the angle and make the same arc. Your two small arcs in the interior of the angle should be crossing.

6. Connect the point where the two small arcs cross to the vertex A of the angle.

You have now created two new angles that are of equal measure (and are each 1/2 the measure of angle BAC.)

Explanation of construction: To understand the explanation, some additional labeling will be needed. Label the point where the arc crosses side AB as D. Label the point where the arc crosses side AC as E. And label the intersection of the two small arcs in the interior as F. Draw segments DF and EF. By the construction, AD = AE (radii of same circle) and DF = EF (arcs of equal length). Of course AF = AF. All of these sets of equal segments are also congruent. We have congruent triangles by SSS. Since the triangles are congruent, any of their leftover corresponding parts are congruent which makes angle BAF equal (or congruent) to angle CAF.



Construct a line through P perpendicular to the given line L.

Directions:

1. Place your compass point on P and sweep an arc of any size that crosses the line twice. You will be creating (at least) a semicircle.

2. Stretch the compass larger.

3. Place the compass point where the arc crossed the line on one side and make a small arc above the line.

4. Without changing the span on the compass, place the compass point where the arc crossed the line on the other side and make another arc. Your two small arcs should be crossing.

5. With your straightedge, connect the intersection of the two small arcs to point P.

This new line is perpendicular to the given line.

Explanation of construction: In this construction, you have bisected the straight angle P. Since a straight angle contains 180 degrees, you have just created two angles of 90 degrees each. Since two right angles have been formed, a perpendicular exists.



Construct a line through a given point P which is parallel to a given line L

Directions:

1 Given a line and a point not on the line P, choose two points A and B on the line on opposite sides of the point. Be sure that the points are not equidistant from the point off the line.

2. Set your compass to the distance between one point on the line and the point off the line. Draw an arc with that radius centered at the other point on the line.

3. Set your compass to the distance between the second point on the line and the point off the line. Draw an arc with that radius centered at first point on the line.

4. Draw a line from the point off the line to the intersection of the two arcs F.

This is your parallel line.



Given line segments of lengths l and l', construct a line segment of length ll'

Directions:

1. Given lengths a and b, place the endpoint of b at any angle greater than zero, at the endpoint of a.

2. Mark off the unit length on a.

3. Draw a segment from the end of the unit segment to the endpoint of b.

4. Construct a parallel line to this segment from the endpoint of a. Extend the segment b out until intersects this line.

The length from the vertex of the angle to the intersection is the length ab.



Given a line segment of length a, construct a line segment of length 1/a

Directions:

1. Given lengths a and 1, place the endpoint of 1 at any angle greater than zero, at the endpoint of a.

2. Mark off the unit length on a.

3. Draw a segment from the end of a to the endpoint of 1.

4. Construct a parallel line to this segment from the endpoint of the unit length.

The length from the vertex of the angle to the intersection is the length 1/a.



Given a line segment of length l, construct a line segment of length square root of L

Directions:

1. Construct segment l+1.

2. Construct a perpendicular bisector of l+1 to find the midpoint.

3. Construct a semi-circle with the center at the midpoint.

4. Construct a perpendicular line at the end of segment l.

The distance from l+1 to the semi-circle is the square root of l