To rotate an equilateral triangle so that it appears exactly the same as its original position, you must rotate it by a specific angle corresponding to its rotational symmetry.
The exact answer is: You have to rotate it 1/3 of the way round, which is 120 degrees.
Understanding Equilateral Triangle Rotation
An equilateral triangle has three equal sides and three equal angles (each 60 degrees). This inherent symmetry allows it to align with its original position after certain rotations around its center point.
As the provided reference states, "You have to rotate it 1/3 of the way round, which is 120 degrees (a full rotation being 360 degrees)." This is the smallest angle of rotation (greater than 0) for which the triangle maps onto itself.
Why 120 Degrees?
A full rotation is 360 degrees. Since an equilateral triangle has three identical vertices and sides arranged symmetrically, it will look the same after rotating by an angle that is a fraction of 360 degrees corresponding to the number of sides or vertices.
- Full Circle: 360 degrees
- Number of Symmetrical Parts: 3
- Rotation for Symmetry: 360 degrees / 3 = 120 degrees
Once you rotate the triangle by this amount, "the three corners have moved around, and the triangle looks the same as before," confirming its rotational symmetry at this angle.
Rotational Symmetry Angles
An equilateral triangle possesses rotational symmetry of order 3. This means it looks identical to its original orientation after rotations of 120 degrees, 240 degrees, and 360 degrees (which brings it back to the starting point).
Here's a simple breakdown:
| Rotation Angle (Degrees) | Appearance Relative to Original |
|---|---|
| 0 | Original Position |
| 120 | Looks the same |
| 240 | Looks the same |
| 360 | Original Position |
By rotating an equilateral triangle by 120 degrees, you effectively cycle the vertices through each position while maintaining the triangle's visual orientation.
