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Geometry - Lines of Symmetry

Grade 4IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A line of symmetry is an imaginary line that passes through the center of a shape or object and divides it into two identical halves. If you fold the shape along this line, both parts will overlap exactly.

An equilateral triangle with a vertical dashed line showing its central axis of symmetry.
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Shapes can have different types of symmetry: vertical, horizontal, or diagonal. A shape like a square possesses all three types.

A square showing four lines of symmetry: one vertical, one horizontal, and two diagonal.
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For regular polygons, the number of lines of symmetry is always equal to the number of sides. For example, a regular pentagon has 55 lines of symmetry.

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Some shapes, like an irregular scalene triangle or a parallelogram (that is not a rhombus), have zero lines of symmetry because no line can divide them into mirror images.

📐Formulae

Number of lines of symmetry in a regular polygon=Number of sides\text{Number of lines of symmetry in a regular polygon} = \text{Number of sides}

Square lines of symmetry=4\text{Square lines of symmetry} = 4

Equilateral triangle lines of symmetry=3\text{Equilateral triangle lines of symmetry} = 3

Circle lines of symmetry=∞ (infinite)\text{Circle lines of symmetry} = \infty\text{ (infinite)}

💡Examples

Problem 1:

Determine how many lines of symmetry a regular hexagon has and describe where they are located.

Solution:

  1. Identify the shape: A regular hexagon has 66 equal sides and 66 equal angles.
  2. Apply the rule: For regular polygons, the number of lines of symmetry equals the number of sides. Therefore, it has 66 lines of symmetry.
  3. Describe locations: 33 lines pass through the opposite vertices (corners), and 33 lines pass through the midpoints of the opposite sides.

Explanation:

Since the hexagon is 'regular,' we use the side-count property. By drawing lines from corner to corner and side-center to side-center, we find all 66 ways to fold the shape into matching halves.

Problem 2:

A student thinks that a standard non-square rectangle has 44 lines of symmetry because they counted the diagonals. Is the student correct? Explain why or why not.

Solution:

  1. Analyze the horizontal and vertical folds: Folding a rectangle vertically or horizontally through the center results in matching halves (22 lines).
  2. Analyze the diagonal fold: If you fold a rectangle along a diagonal, the vertices (corners) do not meet. One corner will stick out past the other.
  3. Conclusion: The student is incorrect. A rectangle has only 22 lines of symmetry.

Explanation:

To be a line of symmetry, the fold must result in a perfect overlap. While a diagonal divides a rectangle into two triangles of equal area, they are not mirror images in their current position, so the diagonal is not a line of symmetry.

Problem 3:

Identify the number of lines of symmetry in an isosceles triangle where two sides are equal.

Isosceles triangle with one vertical line of symmetry.

Solution:

An isosceles triangle has 11 line of symmetry.

Explanation:

In an isosceles triangle, only one line can be drawn from the vertex between the two equal sides to the midpoint of the base. This line creates two mirror-image right-angled triangles.

Problem 4:

Look at the letter 'H'. How many lines of symmetry does it have?

Capital letter H with dashed lines showing vertical and horizontal symmetry.

Solution:

22 lines of symmetry.

Explanation:

The letter 'H' can be split perfectly by a vertical line down the center and a horizontal line across the middle bar.