Geometric Constructions

A geometric construction is a drawing made using only two tools: a compass and an unmarked straightedge. No rulers, no protractors, no measurements at all. You create precise geometric figures purely through the relationships between points, lines, and arcs. This tradition stretches back to the ancient Greeks — Euclid’s Elements (around 300 BCE) formalized the rules — and these same constructions appear in high school geometry courses today.

Why learn constructions when you have rulers and protractors? Because constructions force you to understand why geometry works, not just how to measure. When you bisect an angle with a compass, you are proving — through the act of drawing — that two triangles are congruent. That deeper understanding carries over into proofs, problem solving, and any field that depends on spatial reasoning.

The Two Tools

The Compass

A compass draws arcs and circles. You set it to a width, place the point on the paper, and swing. The critical property: every point on the arc is exactly the same distance from the center. This equal-distance property is the engine behind every construction.

The Straightedge

A straightedge draws straight lines between two points. It has no markings — you cannot measure lengths with it. You can only connect two points or extend a line through two points.

The key rule: You may not transfer a distance by “walking” the compass across the paper while keeping it at the same width (unless a construction explicitly sets the width first). Every distance must be established by placing the compass on specific points.

Construction 1: Copy a Segment

Goal: Given segment $\overline{AB}$, construct a segment of equal length starting at a new point $C$.

Steps:

1. Draw a ray from point $C$ in any direction.
2. Place the compass point on $A$ and open it to reach $B$.
3. Without changing the compass width, place the compass point on $C$ and draw an arc that crosses the ray. Label the intersection $D$.

Segment $\overline{CD}$ is now equal in length to $\overline{AB}$. The compass preserved the distance exactly.

Construction 2: Bisect a Segment (Perpendicular Bisector)

Goal: Given segment $\overline{AB}$, find its midpoint and construct a line perpendicular to $\overline{AB}$ through that midpoint.

Steps:

1. Place the compass point on $A$ and open it to a radius greater than half the length of $\overline{AB}$.
2. Draw an arc above and below the segment.
3. Without changing the compass width, place the compass point on $B$ and draw an arc above and below the segment. The two arcs intersect at two points — call them $P$ (above) and $Q$ (below).
4. Draw the line through $P$ and $Q$. This line is the perpendicular bisector of $\overline{AB}$.

The point where line $PQ$ crosses $\overline{AB}$ is the midpoint $M$ of the segment.

Why it works: Points $P$ and $Q$ are both equidistant from $A$ and $B$ (both are exactly the compass radius away from each endpoint). Any point equidistant from two points lies on the perpendicular bisector of the segment joining them. Since $P$ and $Q$ are two such points, the line through them is the perpendicular bisector.

Construction 3: Copy an Angle

Goal: Given an angle $\angle BAC$, construct a congruent angle at a new vertex.

Steps:

1. Draw a ray from a new point $D$.
2. Place the compass on vertex $A$ and draw an arc crossing both rays of $\angle BAC$. Label these crossings $E$ (on $\overrightarrow{AB}$) and $F$ (on $\overrightarrow{AC}$).
3. Without changing the compass width, place it on $D$ and draw an arc crossing the new ray. Label this crossing $G$.
4. Set the compass width to the distance $EF$.
5. Place the compass on $G$ and draw an arc that intersects the first arc. Label the intersection $H$.
6. Draw ray $\overrightarrow{DH}$. The angle $\angle GDH$ is congruent to $\angle BAC$.

The key idea: you are copying the “opening” of the angle by reproducing the exact chord distance ($EF$) at the same arc radius.

Construction 4: Bisect an Angle

Goal: Given an angle at vertex $V$, construct the ray that divides it into two equal angles.

Steps:

1. Place the compass on vertex $V$ and draw an arc that crosses both rays of the angle. Label the intersection on one ray $P$ and the other $Q$.
2. Place the compass on $P$ and draw an arc in the interior of the angle.
3. Without changing the compass width, place the compass on $Q$ and draw an arc. The two arcs intersect at a point $R$.
4. Draw ray $\overrightarrow{VR}$. This ray bisects the angle at $V$.

Why it works: $P$ and $Q$ are the same distance from vertex $V$ (both lie on the arc of radius $r$). Then $R$ is the same distance from both $P$ and $Q$ (the two arcs had equal radii). So triangle $\triangle VPR \cong \triangle VQR$ by SSS — and therefore $\angle PVR = \angle QVR$.

Construction 5: Construct a Perpendicular Line Through a Point

Goal: Given a line $l$ and a point $P$ on the line, construct a line through $P$ perpendicular to $l$.

Steps:

1. Place the compass on $P$ and draw arcs on both sides of $P$ along the line. Label these intersection points $A$ and $B$. Now $PA = PB$.
2. Increase the compass width. Place the compass on $A$ and draw an arc above (or below) the line.
3. Without changing the width, place the compass on $B$ and draw an arc. The two arcs intersect at $C$.
4. Draw line $\overline{PC}$. This line is perpendicular to $l$ at $P$.

This is essentially the perpendicular bisector construction applied to segment $\overline{AB}$, where $P$ is already the midpoint.

Construction 6: Construct an Equilateral Triangle

Goal: Given segment $\overline{AB}$, construct an equilateral triangle with $\overline{AB}$ as one side.

Steps:

1. Place the compass on $A$ and set the width to the length of $\overline{AB}$. Draw an arc above the segment.
2. Without changing the compass width, place the compass on $B$ and draw an arc above the segment. The two arcs intersect at point $C$.
3. Draw segments $\overline{AC}$ and $\overline{BC}$.

Triangle $\triangle ABC$ is equilateral: $AB = AC = BC$ because each side was created using the same compass width.

Why These Constructions Work

Every compass-and-straightedge construction ultimately relies on one geometric fact: a compass creates points that are all the same distance from a center. This equal-distance property lets you guarantee congruence without measuring.

• Perpendicular bisector: Two points equidistant from both $A$ and $B$ determine the unique line perpendicular to $\overline{AB}$ at its midpoint.
• Angle bisector: Two congruent triangles share a common side (the bisector ray), so the angles on either side of that ray must be equal.
• Equilateral triangle: Three equal-radius arcs force all three sides to be the same length.
• Copying a segment or angle: You physically transfer a distance (or a chord within an arc) to a new location.

The reason these constructions have endured for 2,300 years is that they are proofs you can draw. Each one demonstrates a geometric theorem through the act of constructing it.

Worked Examples

Example 1: Finding a midpoint using perpendicular bisector

Problem: Segment $\overline{AB}$ is 10 cm long. You use the perpendicular bisector construction. What is the length of $\overline{AM}$, where $M$ is the midpoint?

The perpendicular bisector crosses $\overline{AB}$ at its exact midpoint.

$AM = \frac{AB}{2} = \frac{10}{2} = 5 \text{ cm}$

Answer: $AM = 5$ cm.

Example 2: Bisecting a right angle

Problem: You use the angle bisector construction on a $90\degree$ angle. What is the measure of each half?

The bisector divides the angle into two equal parts:

$\frac{90\degree}{2} = 45\degree$

Answer: Each half measures $45\degree$.

Example 3: How many times to bisect $60\degree$ to get $15\degree$?

Problem: Starting with a $60\degree$ angle, how many times must you apply the angle bisector construction to obtain a $15\degree$ angle?

Each bisection halves the angle:

$60\degree \xrightarrow{\text{bisect}} 30\degree \xrightarrow{\text{bisect}} 15\degree$

Answer: Two bisections.

Example 4: Constructing a $90\degree$ angle from a line

Problem: You have a straight line and a point on it. Describe how to construct a $90\degree$ angle using only a compass and straightedge.

Use Construction 5 (Perpendicular through a point on the line):

1. From the point, mark equal distances on both sides along the line.
2. From each of those marks, swing arcs above the line with a larger radius.
3. Connect the point to the arc intersection.

The resulting line makes a $90\degree$ angle with the original line — you have constructed a right angle without a protractor.

Example 5: Constructing a $30\degree$ angle

Problem: Using only a compass and straightedge, describe how to construct a $30\degree$ angle.

Step 1: Construct an equilateral triangle (Construction 6). Each angle is $60\degree$.

Step 2: Bisect one of the $60\degree$ angles (Construction 4).

$\frac{60\degree}{2} = 30\degree$

Answer: Combining the equilateral triangle construction with an angle bisection gives you an exact $30\degree$ angle.

Practice Problems

Test your understanding of geometric constructions. Click to reveal each answer.

Key Takeaways

• Geometric constructions use only two tools: a compass (for equal distances) and an unmarked straightedge (for straight lines)
• The perpendicular bisector construction finds the exact midpoint of a segment and creates a $90\degree$ angle
• The angle bisector construction divides any angle into two equal halves
• You can build many angles by combining constructions — for example, $60\degree$ (equilateral triangle) then bisect to get $30\degree$, or $90\degree$ (perpendicular) then bisect to get $45\degree$
• Every construction works because arcs create equal distances, which guarantee triangle congruence
• These six constructions are the building blocks for nearly all compass-and-straightedge problems in high school geometry

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