Geometry A hub · 01.03

Advanced Constructions

Copy an angle to get a parallel. Two arcs on a line to get a perpendicular. Walk the radius to get a hexagon.

The two symbols

AB ∥ CD means line AB is parallel to line CD. Same-direction arrows on the drawing. They share a plane and never meet.

AB ⊥ CD means perpendicular: they meet at 90°. A square in the corner marks it.

Check a parallel: open the compass to the gap and slide it. Both tips stay on the lines. Check a perpendicular: the corner of a sheet of paper should sit in the angle.

Parallel is a copied angle

You are given a line. You need a parallel through a point off that line. The trick is a transversal: a second line that already cuts the first. Copy the angle at the intersection onto the given point. Corresponding angles congruent ⇒ the new line is parallel.

  1. Draw a line through the given point that cuts the given line. That is the transversal.
  2. Compass on the intersection. Arc that crosses both lines.
  3. Same width, compass on the given point. Matching arc across the transversal.
  4. Open the compass to the chord of the first angle.
  5. Same width on the new arc. Mark the cut. Line through the given point and that cut.
B C
Quiz wording: “a step in constructing a parallel” is often “copy the angle made by the first two lines.” Connecting two arc-crossings with a straightedge is the perpendicular move, not the parallel one.

Perpendicular through a point on the line

Point L is already on the floor line. You want a pole standing at L.

  1. Compass on L. Arc that hits the line on both sides of L. Call the hits D and F.
  2. Open a bit more. Compass on D: arcs above and below.
  3. Same width, compass on F: arcs that cut the first pair.
  4. Line through those two crossings. It runs through L and meets the floor at 90°.

First move: compass at L, arcs on either side of L on the line. Not “two arcs only above the floor.”

L DF

Perpendicular through a point off the line

Point P is above the line (a package in the air). You want the drop to hit the line at 90°.

  1. Compass on P, wider than the distance down to the line.
  2. Swing so you cut the line in two places.
  3. From those two hits, arcs on the far side of the line that meet.
  4. Line from P through that meeting point.

This is the same skeleton as a segment bisector. You are manufacturing two equal lengths on the line, then the perpendicular through the “midpoint.”

Shared step: both “point on the line” and “point off the line” create two arcs (two hits) on the original line. That is the lookalike the quiz asks.

Three polygons on a circle

Inscribed means the vertices sit on the circle. Regular means equal sides and equal angles.

Hexagon and equilateral triangle start the same way: compass set to the radius, walk six ticks around the circle.

  • Connect every tick: regular hexagon. Each side equals the radius.
  • Connect every other tick: equilateral triangle.

A square is a different start. Draw a diameter. Then construct the perpendicular bisector of that diameter (compass a bit wider than the radius, arcs above and below from both endpoints). The two diameters meet the circle in four points. Connect those four.

6 ticks two diameters
Square vs hexagon: square needs a diameter (then a second, perpendicular). Hexagon needs the radius, walked around. Do not swap those tools.

What is this drawing doing?

Partial constructions. Name them before the quiz does.

Practice like 01.03

Ten questions. Next-step on a diagram, same-step in two constructions, square vs hexagon. New items.