Basic Geometric Constructions
A geometric construction creates an exact figure using only a compass and an unmarked straightedge. Equal-radius arcs transfer distances and create congruent segments, congruent angles, midpoints, perpendicular lines, and angle bisectors.
Constructions show why geometric relationships are exact rather than approximate.
Essential Question
How can a compass and straightedge be used to create exact geometric relationships?
Learning Goals
- 1Explain the roles of a compass and unmarked straightedge.
- 2Distinguish a construction from an approximate drawing.
- 3Copy a segment.
- 4Copy an angle.
- 5Construct a perpendicular bisector and midpoint.
- 6Construct an angle bisector.
- 7Construct perpendicular lines through points on or off a line.
Key Vocabulary
Geometric construction
An exact figure created using only a compass and an unmarked straightedge, without measurement.
Compass
A drawing tool with two arms; one holds a pivot point and the other holds a pencil to draw arcs of a fixed radius.
Straightedge
A tool used to draw straight lines; it has no measurement markings.
Construction marks
The arcs and lines drawn during a construction that serve as evidence the construction is valid.
Arc
Part of a circle; in constructions, arcs are drawn with a compass to transfer distances.
Compass width
The fixed distance between the pivot point and the pencil tip of a compass during a construction step.
Copy a segment
A construction that produces a new segment congruent to a given segment by transferring its length with a compass.
Copy an angle
A construction that produces a new angle congruent to a given angle by transferring its arc and chord.
Perpendicular bisector
A line that is perpendicular to a segment and passes through its midpoint.
Midpoint
The point on a segment that divides it into two congruent segments.
Angle bisector
A ray that divides an angle into two congruent angles.
Perpendicular lines
Two lines that intersect to form four right angles (90°).
Key Ideas
A geometric construction creates an exact figure using only a compass and an unmarked straightedge. The compass transfers distances; the straightedge draws lines. Construction marks — the arcs you draw — are evidence that the result is exact, not approximate.
A. Drawing Versus Constructing
- A drawing uses a ruler, protractor, or estimation — it is approximate.
- A construction uses only a compass and unmarked straightedge — it is exact.
- Constructions are valid because they rely on equal radii, not on visual judgment.
- Construction marks must be left visible; erasing them removes the proof.
B. Construction Tools
- The compass transfers exact distances by keeping its width fixed.
- The straightedge draws lines through two points; it does not measure.
- A ruler used as a straightedge is acceptable only if you ignore its markings.
- A protractor is never used in a classical construction.
C. Copying a Segment
- Draw a ray from point C.
- Set the compass width equal to AB (pivot at A, pencil at B).
- Without changing the width, place the pivot at C and mark point D on the ray.
Why it works: The compass width equals AB. Transferring that same width from C produces CD with the same length. CD ≅ AB.
D. Copying an Angle
- Draw a ray from the new vertex V′.
- Draw an arc centered at V (the original vertex) with any radius R; mark intersections E₁ and E₂ on the two rays.
- Draw the same arc (same radius R) centered at V′; mark E₃ on the base ray.
- Set the compass to the chord distance E₁E₂.
- Draw an arc centered at E₃ with that chord distance; mark the intersection P.
- Draw ray V′P.
Why it works: Equal radii create congruent arcs. The chord E₁E₂ encodes the angle measure. Transferring the same chord to the new arc recreates the same angle. ∠V′ ≅ ∠V.
E. Perpendicular Bisector and Midpoint
- Open the compass to more than half the length of AB.
- Draw an arc centered at A (above and below the segment).
- Without changing the compass width, draw an arc centered at B (above and below).
- Label the two intersection points P and Q.
- Draw line PQ. It is the perpendicular bisector of AB.
Why it works: P and Q are each equidistant from A and B (equal radii). Any point equidistant from both endpoints lies on the perpendicular bisector. PQ ⊥ AB at the midpoint M.
F. Constructing an Angle Bisector
- Draw an arc centered at vertex V; mark intersections E₁ and E₂ on the two rays.
- Draw equal-radius arcs centered at E₁ and E₂ (same compass width for both).
- Label the intersection of those two arcs P.
- Draw ray VP.
Why it works: P is equidistant from both sides of the angle (equal radii). Any point equidistant from both sides of an angle lies on the angle bisector. Ray VP bisects the angle.
G. Perpendicular Through a Point on a Line
- Place the compass pivot at P (the given point on the line).
- Draw arcs on both sides of P along the line; mark intersections A and B.
- Open the compass wider (more than PA).
- Draw arcs centered at A and B above the line; mark intersection Q.
- Draw line PQ.
Why it works: A and B are equidistant from P. Q is equidistant from A and B. PQ is the perpendicular bisector of AB, which passes through P. PQ ⊥ line at P.
H. Perpendicular Through a Point off a Line
- Place the compass pivot at P (the given point above the line).
- Draw an arc that crosses the line at two points; label them A and B.
- Draw equal-radius arcs centered at A and B on the opposite side of the line from P; label the intersection Q.
- Draw line PQ.
Why it works: Both P and Q are equidistant from A and B. The line through two points equidistant from A and B is the perpendicular bisector of AB. PQ ⊥ line.
Worked Examples
Goal: Construct segment CD congruent to segment AB.
Given: Segment AB with endpoints A and B.
Tools: Compass, unmarked straightedge.
- Draw a ray starting at point C.
- Place the compass pivot at A and the pencil at B to set the compass width to AB.
- Without changing the compass width, place the pivot at C.
- Draw an arc that crosses the ray; label the intersection D.
Conclusion: CD ≅ AB.
Why it works: The compass width equals AB. Transferring that width from C produces a segment of the same length. No measurement was used — only the compass radius.
Goal: Construct ∠V′ congruent to ∠V.
Given: Angle V formed by two rays.
Tools: Compass, unmarked straightedge.
- Draw a ray from the new vertex V′.
- Draw an arc centered at V with radius R; mark E₁ and E₂ on the two rays.
- Draw the same arc (radius R) centered at V′; mark E₃ on the base ray.
- Set the compass to chord distance E₁E₂.
- Draw an arc centered at E₃ with that chord distance; mark intersection P.
- Draw ray V′P.
Conclusion: ∠V′ ≅ ∠V.
Why it works: Equal radii create congruent arcs. The chord E₁E₂ encodes the angle opening. Transferring the same chord to the new arc recreates the identical angle measure.
Goal: Construct the perpendicular bisector of AB and locate its midpoint M.
Given: Segment AB.
Tools: Compass, unmarked straightedge.
- Open the compass to more than half of AB.
- Draw arcs above and below AB centered at A.
- Without changing the compass width, draw arcs above and below AB centered at B.
- Label the two arc intersections P and Q.
- Draw line PQ. Label its intersection with AB as M.
Conclusion: PQ ⊥ AB at M; M is the midpoint of AB.
Why it works: P and Q are each equidistant from A and B. The locus of points equidistant from two endpoints is the perpendicular bisector. PQ passes through the midpoint and is perpendicular to AB.
Goal: Construct the bisector of ∠V.
Given: Angle V formed by two rays.
Tools: Compass, unmarked straightedge.
- Draw an arc centered at V; mark E₁ on one ray and E₂ on the other.
- Draw equal-radius arcs centered at E₁ and E₂ (same compass width for both).
- Label the intersection of those arcs P.
- Draw ray VP.
Conclusion: Ray VP bisects ∠V; the two resulting angles are congruent.
Why it works: P is equidistant from both sides of the angle (equal radii from E₁ and E₂). The angle bisector is the locus of points equidistant from both sides.
Goal: Construct a line through point P perpendicular to line ℓ, where P is not on ℓ.
Given: Line ℓ and external point P.
Tools: Compass, unmarked straightedge.
- Place the compass pivot at P; draw an arc that crosses ℓ at two points A and B.
- Draw equal-radius arcs centered at A and B on the opposite side of ℓ from P.
- Label the intersection of those arcs Q.
- Draw line PQ.
Conclusion: PQ ⊥ ℓ.
Why it works: P and Q are both equidistant from A and B. The line through two such points is the perpendicular bisector of AB, which is perpendicular to ℓ and passes through P.
Multiple Choice Check
What is the primary purpose of a compass in a geometric construction?
- ATo measure angles in degrees
- BTo draw straight lines between two points
- CTo transfer exact distances by keeping a fixed radius
- DTo verify that two segments are parallel
In the diagram below, a compass arc centered at C has been drawn on a ray. What is the correct next step to copy segment AB?
- AMeasure AB with a ruler and mark the same length from C
- BSet the compass to AB, place the pivot at C, and mark point D where the arc crosses the ray
- CDraw a second ray from C at a 45° angle
- DDraw an arc centered at B with radius BC
The diagram shows a segment with two arcs above and below it, each drawn from both endpoints with the same compass width. The arcs intersect at two points. What construction is being performed?
- ACopying a segment
- BConstructing an angle bisector
- CConstructing the perpendicular bisector
- DCopying an angle
In the diagram, an arc is drawn from vertex V, and two smaller arcs of equal radius are drawn from the arc's intersections with the two rays. The two smaller arcs meet at point P. What do these construction marks indicate?
- ARay VP is perpendicular to one of the original rays
- BRay VP bisects the angle at V into two congruent angles
- CVP is the perpendicular bisector of the arc
- DP is the midpoint of the arc
When copying an angle, why must the compass width be kept the same when drawing the arc on the new vertex as on the original vertex?
- ASo the arc is the same color as the original
- BSo the chord distance can be transferred to create a congruent angle
- CSo the straightedge can be used to measure the angle
- DSo the arc intersects the base ray at a right angle
Guided Practice
Order the steps for copying a segment. Write the correct step number (1–3) next to each action.
A student is copying angle V. After drawing the arc centered at V and marking E₁ and E₂, the student draws the same arc centered at V′ and marks E₃. What is the missing step before drawing ray V′P?
Explain why the compass must be opened to more than half the length of AB when constructing the perpendicular bisector.
In an angle-bisector construction, the arc centered at V intersects the two rays at E₁ and E₂. Equal-radius arcs from E₁ and E₂ intersect at P. What is the final ray of the construction, and what does it do to the angle?
A student constructs a perpendicular from external point P to line ℓ. The arc from P intersects ℓ at A and B. The student then draws arcs from A and B on the same side as P and labels their intersection Q. Is this correct? Explain.
Common Mistakes
Using a marked ruler instead of a straightedge
A ruler with markings introduces measurement, which is not allowed in a classical construction. Use the ruler only as a straightedge and ignore all markings.
Changing the compass width between steps
Once you set the compass width for a step, do not adjust it until that step is complete. Changing the width changes the radius and invalidates the construction.
Measuring distances with a ruler instead of transferring them with a compass
Constructions transfer distances using equal radii. Measuring with a ruler produces an approximation, not an exact construction.
Erasing construction arcs
Construction arcs are the evidence that the construction is valid. Erasing them removes the proof. Always leave arcs visible.
Using unequal arc radii when they must be equal
In the perpendicular bisector and angle bisector constructions, the arcs from both endpoints (or both ray intersections) must have the same radius. Reset the compass if needed.
Setting the compass radius too small for the perpendicular bisector
If the radius is not greater than half of AB, the arcs from A and B will not intersect. Open the compass to more than half the segment length.
Using only one arc intersection point
The perpendicular bisector requires two intersection points (one above and one below the segment) to define the line. One point is not enough.
Drawing the angle bisector through an arbitrary interior point
The bisector must pass through the vertex V and the specific intersection point P found by the construction. An arbitrary interior point does not guarantee equal angles.
Changing the primary arc radius when copying an angle
The arc centered at V and the arc centered at V′ must have the same radius R. If you change the radius, the chord distance will not correspond to the same angle.
Failing to transfer the chord distance when copying an angle
After drawing the matching arc at V′, you must set the compass to the chord E₁E₂ and draw a second arc from E₃. Skipping this step means the angle is not copied.
Trusting visual symmetry instead of construction marks
A line that looks like a bisector or perpendicular may not be exact. Only construction marks — equal-radius arcs and their intersections — guarantee accuracy.
Missing the required given point when constructing a perpendicular
The perpendicular must pass through the specified point (on or off the line). Always verify that your final line passes through that exact point.
Treating construction arcs as part of the final figure
Arcs are construction marks, not part of the final segment, angle, or line. The final figure consists only of the segments and rays drawn with the straightedge.
Measuring from the screen or diagram to verify a construction
Screen measurements are approximate. A construction is verified by its logical steps and equal-radius argument, not by measuring the result.
Before You Finish
- A compass transfers exact distances; it does not measure.
- A straightedge draws lines; it does not measure.
- Construction marks (arcs) are evidence — never erase them.
- Copying a segment preserves the compass width (= the segment length).
- Copying an angle transfers both the arc radius and the chord distance.
- Equal-radius arcs from both endpoints of a segment create the perpendicular bisector.
- Equal-radius arcs from both ray intersections create the angle bisector.
- A perpendicular must pass through the specified point — on or off the line.
Check Your Understanding
A student sets the compass width equal to segment AB, places the compass point at C, and draws an arc to copy the segment. The arc does not intersect the ray from C. What should the student change, and what must remain unchanged?
Explain the difference between a construction and a drawing. Why does the difference matter in geometry?
In a perpendicular-bisector construction, the two arcs from A and B intersect at P above the segment and Q below. A student draws only ray PQ starting from P downward. Is the construction complete? Explain.