TwiddleLab / Math / Shapes and Proof

Constructions

Lesson 9 of 10 · 7 min · Free, no account needed

What you will learn

• A construction uses only a compass (equal distances) and a straight edge (lines through two points).
• Two equal circles from A and B cross on the perpendicular bisector of AB: it halves AB at a right angle.
• Every point on it is equally far from A and B. That's also the border between two nearest neighbours.

Start the lesson

How the lesson goes

1. No ruler markings, no protractor. Can you still cut a segment exactly in half?

   First, by eye.

2. Open the compass.

   Same opening at A and at B. Find an opening where the two circles cross at two points, then Draw.

3. Now the straight edge.

   Join the two crossing points with a line.

4. What's special about that line?

   Drag X to three spots on the blue line, and once off it. Compare XA and XB.

5. Names for what you built.

   A construction uses only a compass and a straight edge. No numbers, yet exact.

   The line through the two crossings is the perpendicular bisector of AB: it cuts AB in half, at 90°.

   Why it works: each crossing point is one opening from A and one from B. Every point on the line is equally far from A and B.

6. Halve an angle the same way.

   Tap the points in the right order: corner, a mark, the other mark, the crossing.

7. Two stores. Every customer goes to the nearer one. Where is the border?

   Drag the grey customer. Drop three flags where the two stores are equally near.

8. Game: a hexagon in four moves.

   The compass is open one radius. Swing arcs from marked points until there are 6 corners. Par is 4 moves.

9. Your constructions, as a film.

Previous lesson: Coordinates Meet Geometry · Next lesson: Project: Prove It Yourself · Part of the course: Shapes and Proof

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