Standard Geometric Constructions: Bisectors and Perpendiculars

• First, place the compass point on endpoint $A$ of the line segment. Open the compass so the radius is clearly more than half the length of the segment.
• Note that if the radius is $\leq$ half the length, the arcs will not intersect, and the construction will fail.
• Next, draw a large arc above and below the line.
• Then, keeping the compass width absolutely identical, place the point on endpoint $B$ and draw construction arcs to intersect the first set of arcs.
• Finally, use a ruler to draw a straight line directly through the two points where the arcs intersect.

This newly drawn line acts as the locus of points that are equidistant from both endpoints of the original segment.

Constructing a Perpendicular to a Given Line

What is the absolute shortest path you can take to walk from a point in a field directly to a straight road? The shortest distance from a point to a line is always a perpendicular path. In construction exams, using a protractor to draw this $90^{\circ}$ line will result in zero marks; you must only use a ruler and compasses.

From a point $P$ OFF the line $L$:

• First, place the compass on point $P$ and open it wider than the distance to the line. Draw an arc that crosses line $L$ at two distinct points, $A$ and $B$.
• Next, place the compass on point $A$ and draw an arc on the opposite side of the line from $P$.
• Then, using the exact same compass width, place the point on $B$ and draw an arc intersecting the previous one at point $Q$.
• Finally, draw a straight line through $P$ and $Q$.

At a point $P$ ON the line $L$:

• First, place the compass on point $P$ and draw two arcs crossing line $L$ on either side, creating points $A$ and $B$ that are an equal distance from $P$.
• Next, open the compass slightly wider (approximately three-quarters of the way to the opposite point).
• Then, place the compass on $A$ and draw an arc above the line. Using the same width, place it on $B$ and draw an intersecting arc at point $Q$.
• Finally, draw a straight line through $Q$ and $P$.

Bisecting a Given Angle

You can easily fold a piece of paper in half to divide an angle perfectly, but you cannot fold an exam paper to find the exact center of drawn lines. Instead, you must construct an angle bisector, which represents the locus of points equidistant from the two arms of the angle.

• First, place the compass point on the vertex (point $B$). Draw an arc that crosses the first arm $AB$ at point $X$ and the second arm $BC$ at point $Y$.
• Next, place the compass point exactly on intersection $X$ and draw an arc in the central area between the two arms.
• Then, without changing the compass width, place the point on intersection $Y$ and draw an arc that crosses the one from the previous step at point $P$.
• Finally, use a ruler to draw a straight line from the vertex $B$ directly through intersection $P$.

This construction accurately splits the original angle into two congruent angles. We can represent this mathematically as:

$$\angle ABP = \angle CBP = \frac{1}{2} \angle ABC$$

If the angle you are bisecting is very small, use a larger compass setting for the final intersecting arcs to improve precision and ensure the arcs cross cleanly.