Jacobs Geometry
Contents 8 sections

Lesson 4 — Constructions: Telling Time with Shadows

Big idea: A construction is a drawing made with only a straightedge and compass. Equal radii create equal lengths.

Words to know

Word Meaning
Straightedge An unmarked ruler; used to draw lines but not to measure.
Compass A tool for drawing circles and arcs.
Circle The set of all points in a plane a fixed distance from a given point.
Center The given point in the middle of a circle.
Radius The fixed distance from the center to the circle.
Construction A drawing made using only a straightedge and compass.
Bisect To divide into two equal parts.

Concepts to know

  • A straightedge draws straight lines; a compass draws circles or parts of circles called arcs.
  • A construction is different from a careful drawing: it is justified by geometry, not by measuring.
  • To bisect a segment or angle is to cut it into two equal parts.
  • The two basic constructions are:
    1. bisect a line segment;
    2. bisect an angle.
  • The equal compass settings create equal lengths; those lengths form congruent triangles.

Construction 1 — Bisect a line segment

  1. Open the compass to more than half of AB.
  2. With A as center, draw arcs above and below the segment.
  3. With B as center and the same radius, draw arcs crossing the first arcs at C and D.
  4. Draw CD. It crosses AB at the midpoint M, so AM = MB.

Construction 2 — Bisect an angle

  1. With the vertex C as center, draw an arc crossing both sides at A and B.
  2. With A as center, draw an arc inside the angle.
  3. With B as center and the same radius, draw an arc crossing the previous arc at D.
  4. Draw CD, so ∠ACD = ∠DCB.

Homework helper

If the homework asks… Do this…
Bisect a segment Use Construction 1; the crossing points give the perpendicular bisector.
Bisect an angle Use Construction 2; start at the vertex.
Check a bisection Measure both parts; they should be equal.
Explain why it works Point to the equal radii and the congruent triangles they create.
Find north by shadows Bisect the angle between the two equal shadow positions.
Draw a construction Keep the compass radius fixed and make clear arcs.

Worked example

To bisect segment AB:

  • Draw equal arcs from A and from B.
  • Label the two crossing points C and D.
  • Draw CD.

The point where CD crosses AB is the midpoint M, so AM = MB.

Watch out

  • Do not change the compass radius in the middle of a construction.
  • Draw arcs long enough to cross clearly.
  • Do not erase construction marks; they are part of the work.
  • The straightedge has no measurements; do not use it like a ruler.
  • An angle bisector must pass through the vertex.

Before you turn it in