Basic Geometric Constructions

In geometry, a construction is the process of drawing shapes, angles, or lines accurately using only two tools: a compass (to draw circles or arcs) and a straightedge (an unmarked ruler to draw straight lines). No measuring tools like rulers or protractors are allowed.

Here is a guide to the fundamental geometric constructions.

1. Copying a Line Segment

To construct a segment congruent to a given segment $AB$:

1. Use a straightedge to draw a long working line. Mark a starting point $C$ on this line.
2. Place the point of the compass on $A$ and adjust the pencil so it exactly touches $B$. This sets the compass width to the length of $AB$.
3. Without changing the compass width, place the compass point on $C$ and draw an arc that intersects the working line.
4. Label the intersection $D$. The segment $CD$ is exactly the same length as $AB$.

2. Constructing a Perpendicular Bisector

A perpendicular bisector cuts a line segment exactly in half at a $90^\circ$ angle.

1. Start with a line segment $AB$.
2. Place the compass point on $A$ and open it to a width that is clearly greater than half the length of $AB$.
3. Draw a large arc above and below the segment.
4. Keeping the exact same compass width, place the compass point on $B$ and draw another arc that intersects the first arc in two places (one above the segment, one below).
5. Use a straightedge to draw a line connecting the two intersection points. This line is the perpendicular bisector of $AB$.

3. Copying an Angle

To construct an angle congruent to a given $\angle A$:

1. Draw a straight working line with a starting point $A'$.
2. Place the compass point on the original vertex $A$ and draw an arc that intersects both sides of $\angle A$. Label these intersections $B$ and $C$.
3. Keeping the same compass width, place the point on $A'$ and draw a similar arc across the working line. Label the intersection $B'$.
4. Go back to the original angle. Place the compass point on $B$ and adjust the pencil to touch $C$. This measures the "width" of the angle.
5. Move the compass to $B'$, and draw an arc that intersects the arc you drew in Step 3. Label this intersection $C'$.
6. Draw a ray from $A'$ through $C'$. The new angle $\angle A'$ is congruent to $\angle A$.

4. Bisecting an Angle

An angle bisector divides an angle into two equal smaller angles.

1. Place the compass point on the vertex of the angle.
2. Draw an arc that intersects both rays of the angle. Label these intersection points $X$ and $Y$.
3. Place the compass point on $X$ and draw an arc in the interior of the angle.
4. Keeping the same compass width, place the point on $Y$ and draw another arc that intersects the previous one. Label the intersection $Z$.
5. Use a straightedge to draw a ray from the original vertex through $Z$. This ray bisects the angle.