Jacobs Geometry
Contents 8 sections

Lesson 7 — Constructions

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

Words to know

Word Meaning
Construction A geometric drawing made using only a straightedge and compass.
Straightedge An unmarked ruler; draws lines but does not measure.
Compass A tool for drawing circles and arcs.
Arc Part of a circle.
Radius The distance from the center of a circle to the circle.
Copy Reproduce a segment, angle, or triangle exactly.
Bisect Divide into two equal parts.

The five constructions

# Construction Quick steps
1 Bisect a segment Equal arcs from both endpoints; draw the line through the two intersections.
2 Bisect an angle Arc across both sides; equal arcs from those points; draw from the vertex.
3 Copy a segment Set the compass to the segment; mark that length on a new line.
4 Copy an angle Draw an arc; copy the same arc; transfer the chord distance; draw the ray.
5 Copy a triangle Copy one side; draw arcs using the other two side lengths; connect.

Concepts to know

  • Constructions use only a straightedge and compass.
  • The compass radius must stay fixed during a construction unless the steps say to change it.
  • Construction 4 works because the copied arcs create two triangles with all three sides equal (SSS). The corresponding angles are then equal.
  • Construction 5 works because the three copied side lengths determine the triangle (SSS).
  • Constructions 1 and 2 also depend on congruent triangles.

Construction 4 — Copy an angle

  1. With the vertex A as center, draw an arc crossing both sides at D and E.
  2. Draw ray BC as one side of the new angle.
  3. With B as center and the same radius, draw an arc crossing BC at F.
  4. Set the compass to the distance DE.
  5. With F as center, draw an arc crossing the previous arc at G.
  6. Draw BG.

Then ∠A = ∠B.

Why: Draw DE and FG. ΔADE ≅ ΔBFG by SSS, so ∠A = ∠B.

Construction 5 — Copy a triangle

  1. Draw a line and copy side AB onto it as DE.
  2. Set the compass to AC. With D as center, draw an arc.
  3. Set the compass to BC. With E as center, draw an arc crossing the first at F.
  4. Draw DF and EF.

Then ΔABC ≅ ΔDEF by SSS.

Homework helper

If the homework asks… Do this…
Copy a segment Use Construction 3.
Copy an angle Use Construction 4; keep the first arc radius the same.
Copy a triangle Use Construction 5; copy one side, then arc the other two lengths.
Bisect a segment or angle Use Construction 1 or 2.
Check your construction Measure with a ruler or protractor; the parts should be equal.
Explain why it works Name the congruent triangles and the SSS reason.

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.
  • A straightedge is not a ruler; do not use its markings to measure.
  • Label the corresponding points in the same order.

Before you turn it in