Construct 60° from an equilateral triangle, halve it to 30° and 15°, and build hexagons, stars and designs.
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Two arcs of the same radius make an equilateral triangle, and 60°.
We need a 60° angle at point A on a segment AX.
Then ∠CAX = 60°.
Chaining constructions: 60°, then halves, then a hexagon.
Problem. Construct 60°, then 30° and 15°, then a regular hexagon of side 5 cm. (Textbook pp. 153–154)
Choose how to make each angle.
How do you construct 30°?
A six-pointed star hides a hexagon and six equilateral triangles.
This star has rotational symmetry. The book's hint: A, B, C, D, E, F are the star's tips, and G, H, I, J, K, L are its inner corners.
Build a hexagon, then a triangle on each side, and check it is equilateral.
Problem. Construct the six-pointed star, and show that each point is an equilateral triangle. (Textbook p. 154)
The book's designs and the optical illusion, for the next task.
Q1. Designs (Textbook pp. 154–155). (b) a 6-petal flower that can be made with a compass alone:
Build the book's designs, a star in a hexagon, and explain an illusion.
Figure it Out, Textbook pp. 154–155. Use the figures shown above.
Q1. Construct these figures: (a) the inflexed arc in your textbook (p. 154); (b) the 6-petal flower, with a compass alone; (c) the hexagon in a circle; (d) the ring of six circles; (e) the hexagon covered with a triangle grid.
Q2. Optical illusion. What do you notice in the figure? How does it happen? Recreate it.
Q3. Construct the star in a hexagon. Hint: find the angles.
Name the angles and radii.
What is actually drawn?
Angles at the hexagon's corners and at the star's inner corners.