Construct perpendicular bisectors and angle bisectors using a compass, and explore how shapes tile a plane without gaps.
A perpendicular bisector of a line segment crosses it at exactly the midpoint, at a right angle. An angle bisector splits an angle into two exactly equal halves. Both are constructed precisely using a compass, without measuring.
When constructing a perpendicular bisector using a compass, why must the arcs above and below the segment use the same radius?
So the two intersection points are equally distant from both endpoints, ensuring the resulting line passes through the exact midpoint at 90°
An angle bisector is constructed for a 60° angle. What are the two resulting angles?
30° and 30°, since the bisector splits it exactly in half
A tiling (or tessellation) covers a flat surface completely with shapes, leaving no gaps and no overlaps. Certain shapes, like squares, equilateral triangles, and hexagons, tile perfectly on their own because their angles divide evenly into 360°.
Can regular hexagons tile a plane on their own? Why?
Yes, because each interior angle is 120°, and 3 hexagons meeting at a point give 3×120° = 360°
Can regular pentagons tile a plane on their own?
No, because each interior angle is 108°, and no whole number of pentagons meeting at a point adds up to exactly 360°
1. An angle bisector is constructed for a 90° angle. What are the two resulting angles?
2. An angle bisector is constructed for a 100° angle. What are the two resulting angles?
3. Can equilateral triangles tile a plane on their own? Why?
4. Can regular octagons tile a plane on their own?
5. Why is a perpendicular bisector useful for finding the exact midpoint of a segment without measuring?