Axes of Symmetries of Shapes (Leaving Cert Mathematics): Revision Notes

Axes of Symmetries of Shapes

Overview

Symmetry is a fundamental concept in geometry that describes a shape's ability to remain unchanged under certain transformations, such as reflexion. Axial symmetry refers to symmetry around a line, called the axis of symmetry. The axes of symmetry depend on the shape's geometry.

---

Common Shapes and Their Axes of Symmetry

Circle

• Axes of Symmetry: Infinite axes of symmetry, as every diameter acts as an axis of symmetry.

Equilateral Triangle

• Axes of Symmetry: 3.
• Each axis passes through a vertex and the midpoint of the opposite side.

Square

• Axes of Symmetry: 4.
• Two axes pass through the midpoints of opposite sides, and two pass through opposite vertices (diagonals).

Rectangle

• Axes of Symmetry: 2.
• Both axes pass through the midpoints of opposite sides.

Rhombus

• Axes of Symmetry: 2.
• Both axes are the diagonals of the rhombus.

Regular Pentagon

• Axes of Symmetry: 5.
• Each axis passes through a vertex and the midpoint of the opposite side.

Regular Hexagon

• Axes of Symmetry: 6.
• Three axes pass through opposite vertices, and three pass through the midpoints of opposite sides.

Parallelogram (Non-Rectangle)

• Axes of Symmetry: None, as opposite sides are parallel but not symmetrical about any axis.

Isosceles Triangle

• Axes of Symmetry: 1. The axis passes through the vertex and the midpoint of the base.

Kite

• Axes of Symmetry: 1. The axis passes through the two distinct vertices.

---

Applications of Axial Symmetry

• Reflection Symmetry: Used in geometry to determine congruency and balance in shapes.
• Design and Art: Symmetry contributes to aesthetics in designs and patterns.
• Mathematics and Physics: Helps in simplifying problems involving geometric shapes.

---

Summary

• Axial Symmetry: Describes symmetry around a line, known as the axis of symmetry.
• Key Shapes and Symmetries:
  • Circle: Infinite axes.
  • Equilateral Triangle: 3 axes.
  • Square: 4 axes.
  • Rectangle: 2 axes.
  • Regular Hexagon: 6 axes.
  • Parallelogram: No axes.
• Axial symmetry is crucial in understanding the geometric properties of shapes.