Question

Describe any rotations less than or equal to 360° that map the polygon onto itself.

5. regular octagon
6. equilateral triangle

Answer 1

A regular octagon can be rotated by multiples of 45°, such as 90° and 180°, to map onto itself, while an equilateral triangle maps onto itself with rotations of 120° and 240°.

Step-by-step explanation:

The question concerns geometric transformations such as rotations that map a polygon onto itself. A regular octagon and an equilateral triangle, being highly symmetrical shapes, exhibit this property.

• Regular Octagon: For a regular octagon, with its eight equal sides and eight equal angles, any rotation by a multiple of 45° (up to 360°) will map the octagon onto itself. This includes the rotations of 45°, 90°, 135°, 180°, 225°, 270°, and 315°.

• Equilateral Triangle: An equilateral triangle, which has three equal sides and three equal angles of 60° each, will map onto itself through rotations of 120° and 240°, as well as a full rotation of 360° (which is the same as no rotation).