Consider the regular do-decagon shown below: (a) How many lines of symmetry does the regular do-decagon have? Draw all of the lines of symmetry in the figure above. (b) Determine the angle of rotational symmetry for the regular do-decagon.
The Correct Answer and Explanation is:
Let’s break down your questions step by step:
(a) Number of Lines of Symmetry in a Regular Do-Decagon
A regular do-decagon has 12 sides, and for any regular polygon, the number of lines of symmetry is equal to the number of sides. Thus, a regular do-decagon has 12 lines of symmetry. These lines can be drawn in two different ways:
- Through opposite vertices: These lines will pass through pairs of opposite vertices, dividing the decagon into two symmetrical halves.
- Through the midpoints of opposite sides: These lines will pass through the midpoints of opposite sides, dividing the decagon in a way that reflects each side into its counterpart.
So, you can draw 12 lines of symmetry in total, ensuring each one is either through opposite vertices or through the midpoints of opposite sides.
(b) Angle of Rotational Symmetry for the Regular Do-Decagon
The angle of rotational symmetry is the smallest angle by which the figure can be rotated such that it looks the same as it did before the rotation.
For a regular do-decagon, which has 12 sides, the angle of rotational symmetry is calculated by dividing 360 degrees by the number of sides. This gives us:Angle of rotational symmetry=360∘12=30∘\text{Angle of rotational symmetry} = \frac{360^\circ}{12} = 30^\circAngle of rotational symmetry=12360∘=30∘
Therefore, the regular do-decagon has a rotational symmetry of 30 degrees. This means that rotating the do-decagon by multiples of 30 degrees (e.g., 30°, 60°, 90°, etc.) will map the shape onto itself.
Summary:
- (a) The regular do-decagon has 12 lines of symmetry.
- (b) The angle of rotational symmetry is 30°.
