Exercises: Make Formal Geometric Constructions

It measures the length of a segment in centimeters.

It draws all points at a fixed distance from a center point.

It draws a straight line through two given points.

It measures the size of an angle in degrees.

The line through A and B

The perpendicular bisector of segment AB

A line parallel to AB

The angle bisector of angle A

ASA (Angle-Side-Angle)

AAS (Angle-Angle-Side)

SSS (Side-Side-Side)

SAS (Side-Angle-Side)

Use a ruler to measure AB in millimeters, then mark the same length from P.

Set the compass width to AB by placing the point at A and pencil at B, then draw an arc from P without changing the width.

Draw a ray from P and estimate where to mark the endpoint by eye.

Place the straightedge so it touches both A and P, then mark where it meets the arc.

P and Q were drawn by hand to look symmetrical, so the line between them appears perpendicular.

P and Q are each equidistant from A and B, so both lie on the perpendicular bisector of AB by the equidistance theorem.

The arcs cross exactly at the midpoint of AB, so the line through the crossing points is perpendicular.

A and B are the same distance from the line PQ, which means PQ must be vertical.

Yes — setting the radius to exactly half ensures the arcs meet at the midpoint.

No — setting the radius to exactly half means the arcs only just touch at the midpoint and do not produce two distinct intersection points needed to draw the bisector.

No — the radius must be set to exactly the full length of AB for the construction to work.

Yes — the compass radius does not affect where the arcs intersect, so any setting works.

Copy the angle between the transversal and line l at their intersection point Q, placing the congruent angle at P on the same side of the transversal — this creates equal corresponding angles, guaranteeing parallelism.

Copy the right angle (90 degrees) at the intersection of the transversal and l, so the new line is perpendicular to the transversal.

Copy the angle between the transversal and l at Q, placing it on the opposite side of the transversal at P — this creates alternate interior angles.

Draw a line through P that appears to run alongside l — no angle copying is needed because parallel lines look equidistant.

Copying angle BAC to a new vertex

Bisecting angle BAC

Constructing a perpendicular to ray AB at point A

Constructing a line parallel to ray AC through point D

No — the midpoint depends on the compass radius, so different settings produce different midpoints.

Yes — the midpoint is a fixed geometric property of the segment, and any radius greater than half of AB produces the correct perpendicular bisector through it.

No — Priya's arcs will be larger and therefore cross the segment at a different location.

Yes — but only because 3 cm and 5 cm happen to give the same arc intersection points.

Using a ruler to draw a segment 7 cm long, then measuring 3.5 cm from one end to mark the midpoint.

Using a protractor to draw a 60-degree angle by aligning it with a baseline and marking the degree.

Using compass arcs from both endpoints to find two points equidistant from the endpoints, then connecting them with a straightedge to find the midpoint.

Folding a piece of paper at what looks like the midpoint, then pressing the fold to mark it.

The construction will not work on CD because it was designed only for short segments.

The student must use a different method for CD because the compass width will need to exceed half of CD, which may be physically difficult.

The same construction works on CD. The student simply needs a compass radius greater than half of CD, and the result will correctly bisect CD.

The construction gives an approximate midpoint for very long segments because compass arcs become less accurate over large distances.

Draw a line through P that appears equidistant from l at both ends.

Drop a perpendicular from P to l, then construct a perpendicular to that perpendicular at P.

Measure the distance from P to l with a ruler, then mark points at the same distance on both sides of P.

Draw any arc centered at P and place a point on l; connect P to that point.

Because the Mira is a straightedge, and any line through the midpoint is perpendicular to the segment.

Because the Mira acts as a mirror (line of reflection), and reflecting C onto D means every point on the Mira is equidistant from C and D — satisfying the definition of the perpendicular bisector.

Because the Mira physically measures the distance from C to D and marks the midpoint at half that distance.

Because the Mira produces a perpendicular line by default whenever placed on a segment.

Jordan should have measured in millimeters, not centimeters, for greater precision.

Jordan performed a drawing, not a formal construction. Ruler measurements always contain some physical error, and the validity of the result depends on the accuracy of the tool, not geometric logic.

Jordan copied the segment correctly but should also have verified the result with a protractor.

Jordan's method only works if PQ is parallel to AB.

Error 1: She should use a ruler, not a compass. Error 2: She should draw only one circle, not two.

Error 1: Setting the radius to exactly half of AB means the arcs touch at only one point (on the segment itself), giving no second reference point. Error 2: The bisector construction requires two intersection points to define the perpendicular bisector line — one point is not enough.

Error 1: The compass radius should equal the full length of AB. Error 2: She should not draw complete circles — only short arcs.

Error 1: She drew circles above and below the segment when arcs should intersect on the segment. Error 2: One intersection point is sufficient if it is on the midpoint.