8.4 Geometric Constructions with Compass, Straightedge & Reflection Devices
Key Takeaways
A construction uses only a compass and an unmarked straightedge (or a reflection device or paper folding), never a ruler's scale or a protractor.
The perpendicular bisector construction works because each arc intersection is equidistant from the segment's endpoints, and such points lie on the perpendicular bisector.
The angle bisector and copy-an-angle constructions are justified by SSS triangle congruence followed by CPCTC.
A reflection device's edge is a line of reflection, so placing it so that A reflects onto A′ draws the perpendicular bisector of AA′.
The circumcenter (perpendicular bisectors) is equidistant from a triangle's vertices, while the incenter (angle bisectors) is equidistant from its sides.
8.4 Geometric Constructions with Compass, Straightedge and Reflection Devices
Two competencies name constructions directly. Competency 009 asks you to "describe and justify geometric constructions made using a compass and straightedge and other appropriate technologies." Competency 011 asks the same for constructions made with a reflection device. On the exam, questions usually give a picture of construction marks and ask what was constructed or why the construction works. The justification almost always rests on congruent triangles or on the properties of a rhombus.
What Counts as a Construction
A construction creates a figure using only idealized tools, with no measuring:
- A straightedge (an unmarked ruler) draws the line through two points.
- A compass draws a circle with a given center through a given point. In practice it also copies a distance.
- A reflection device is a semi-transparent mirror, often sold as a Mira. You can see the reflection of a figure and trace it on the far side. The edge of the device acts as a line of reflection.
- Paper folding and dynamic geometry software perform the same constructions, and software also lets students drag the figure to test that the construction still holds.
These tools echo Euclid's first three postulates: draw a segment between two points, extend it, and draw a circle with any center and radius.
The Core Compass-and-Straightedge Constructions
| Construction | Key steps | Why it works |
|---|---|---|
| Copy a segment | Set the compass to AB and mark that length on a ray | The compass preserves distance |
| Copy an angle | Draw an arc across both sides of the given angle, draw the same arc from the new vertex, then copy the chord width | SSS: the two triangles formed by the arcs are congruent, so the angles are congruent |
| Perpendicular bisector of AB | With radius more than half of AB, draw arcs from A and from B. Join the two intersection points. | Each intersection point is equidistant from A and B, and such points lie on the perpendicular bisector. The four points also form a rhombus, whose diagonals are perpendicular bisectors of each other. |
| Angle bisector | From the vertex, draw an arc cutting both sides. From those two points draw equal arcs. Join the vertex to their intersection. | SSS gives two congruent triangles, so the two angles at the vertex are congruent |
| Perpendicular through a point P on line ℓ | Mark two points on ℓ equidistant from P, then bisect that segment | It is the perpendicular bisector of a segment centered at P |
| Perpendicular from a point P not on ℓ | Draw an arc from P cutting ℓ twice, then construct the perpendicular bisector of that chord | P is equidistant from the two cut points, so it lies on their perpendicular bisector |
| Parallel through a point P | Draw a transversal through P and copy the angle it makes with ℓ at P as a corresponding angle | Corresponding Angles Converse |
| Equilateral triangle on AB | Draw circles centered at A and B with radius AB, and join A and B to an intersection point | All three sides equal the radius |
| Regular hexagon in a circle | Step off the radius six times around the circle | Each chord equals the radius, so there are six equilateral triangles |
| Square in a circle | Draw a diameter, construct the perpendicular diameter, and join the endpoints | Congruent diagonals that are perpendicular bisectors of each other make a square |
Points of concurrency (triangle centers)
| Center | Constructed from | Special property |
|---|---|---|
| Circumcenter | Perpendicular bisectors of the sides | Equidistant from all three vertices, so it is the center of the circumscribed circle. It lies outside an obtuse triangle and at the midpoint of the hypotenuse of a right triangle. |
| Incenter | Angle bisectors | Equidistant from all three sides, so it is the center of the inscribed circle. It is always inside the triangle. |
| Centroid | Medians (a vertex to the midpoint of the opposite side) | The balance point. It divides each median in a 2 : 1 ratio from the vertex. |
| Orthocenter | Altitudes | Where the altitudes meet. It lies at the right-angle vertex of a right triangle. |
Application example. Three schools sit at points A, B and C. A district wants a bus depot equally far from all three schools. The depot belongs at the circumcenter, where the perpendicular bisectors meet. If the district instead wants a sports field equally far from three straight roads that form a triangle, it belongs at the incenter.
Constructions with a Reflection Device
Because the device's edge is a line of reflection, each construction rests on one fact: a line of reflection is the perpendicular bisector of every segment joining a point to its image.
- Reflect a figure. Place the device on the given line and trace the image you see.
- Find the line of reflection between a figure and its image. Move the device until the reflection of the figure lies exactly on the image, then draw along the edge.
- Perpendicular bisector of AB. Position the device so the reflection of A falls exactly on B. The edge is the perpendicular bisector.
- Angle bisector. Place the device at the vertex so the reflection of one side lies on the other side. The edge bisects the angle.
- Perpendicular from a point to a line. Keep the line reflecting onto itself while the edge passes through the point.
- Lines of symmetry. Any placement where the reflection of a figure matches the figure itself marks a line of symmetry.
Paper folding does the same work. Folding A onto B creates the perpendicular bisector, and folding one side of an angle onto the other creates the angle bisector.
Justifying a Construction: A Model Argument
Claim: The compass-and-straightedge angle bisector construction produces a ray that bisects .
- The first arc centered at meets the sides at and , so (equal radii).
- Equal arcs from and meet at , so (equal radii).
- (reflexive property).
- by SSS.
- by CPCTC, so ray bisects .
Almost every construction proof follows this pattern: equal compass settings produce equal segments, the segments form congruent triangles, and CPCTC gives the conclusion.
Three Famous Impossibilities
The ancient Greeks asked whether compass and straightedge could trisect an arbitrary angle, double the cube (construct the edge of a cube with twice the volume), or square the circle (construct a square with the same area as a given circle). In the 1800s, mathematicians proved all three impossible with these tools. The proofs use algebra that connects constructible lengths with certain kinds of numbers, and the circle problem fails because is transcendental. This history illustrates Standard VI: mathematical questions can stay open for centuries and be settled by ideas from another branch of mathematics.
Teaching Notes
- Measuring is not constructing. A student who uses a protractor to "bisect" an angle has drawn a figure, not constructed one. Ask students to explain why each arc guarantees equal lengths.
- Keep compass settings consistent. Many failed perpendicular bisectors come from changing the radius between the arc from A and the arc from B.
- Angle bisector versus median. In a scalene triangle, the bisector of an angle does not usually pass through the midpoint of the opposite side. Constructing both on the same triangle makes the difference visible.
- Connect to transformations. A reflection device makes the link between perpendicular bisectors and reflections concrete, which prepares students for defining congruence through rigid motions.
A student draws arcs of equal radius, each more than half of AB, centered at A and at B. The arcs meet at points P and Q, and the student draws line PQ. Which statement best justifies that line PQ is the perpendicular bisector of AB?
P and Q were placed by estimating the midpoint of AB, so PQ passes through it
The arcs have equal radii, so PA = PB and QA = QB; points equidistant from A and B lie on the perpendicular bisector of AB
Any two arcs drawn from A and B always meet on a line parallel to AB
The construction works because the compass measured AB and divided it by 2
Using a reflection device, a student adjusts its position until the reflection of point A lands exactly on point A′, then draws a line along the edge of the device. What has the student constructed?
A line parallel to segment AA′ through its midpoint
The angle bisector of the angle formed at A′
A line through A′ that makes a 45° angle with AA′
The perpendicular bisector of segment AA′
Three towns are located at the vertices of an obtuse triangle. Planners want a radio tower that is the same distance from all three towns. Where should the tower be placed, and how is that point constructed?
At the incenter, found by intersecting the angle bisectors; it is always inside the triangle
At the centroid, found by intersecting the medians; it divides each median 2 : 1
At the circumcenter, found by intersecting the perpendicular bisectors of the sides; for an obtuse triangle it lies outside the triangle
At the midpoint of the longest side, because that point is always equidistant from all three vertices
Sections you finish are checked off in the contents.