Review the key concepts, formulae, and examples before starting your quiz.
🔑Concepts
A line of symmetry is an imaginary line that divides a shape into two identical parts. If you fold a shape along this line, the two halves will overlap perfectly, appearing as mirror images of each other. Visually, imagine placing a small mirror on the line; the reflection should complete the shape exactly as it looks in real life.
Lines of symmetry can be oriented in different directions: vertical (running straight up and down), horizontal (running left to right), or diagonal (slanted at an angle). For example, a human face has a vertical line of symmetry down the middle, while a drawing of a boat on water might have a horizontal line of symmetry at the water level.
Regular polygons are shapes where all sides are the same length and all angles are equal. A key rule is that a regular polygon has the same number of lines of symmetry as it has sides. For instance, an equilateral triangle has lines of symmetry, each passing through a vertex and the midpoint of the opposite side.
In quadrilaterals, the number of lines depends on the properties of the sides and angles. A square has lines of symmetry (one vertical, one horizontal, and two diagonal), whereas a rectangle only has (one vertical and one horizontal). It is important to note that the diagonals of a non-square rectangle are not lines of symmetry because the corners do not match up when folded.
Circles are special geometric figures because they possess an infinite number of lines of symmetry. Any straight line that passes through the center point of the circle acts as a line of symmetry, dividing the circle into two identical semicircles.
Many letters of the alphabet and numbers also exhibit symmetry. For example, the letter has one vertical line of symmetry down the center, the letter has both a vertical and a horizontal line, and the letter has no lines of symmetry because it cannot be folded to match its halves perfectly.
Symmetry is often found in nature and architecture. A butterfly exhibits symmetry with its wings, where a vertical line drawn through its body shows that the left wing is a reflection of the right wing. This is known as bilateral symmetry.
📐Formulae
💡Examples
Problem 1:
Determine how many lines of symmetry a regular hexagon has and describe where they are located.
Solution:
- Identify the shape: A regular hexagon has equal sides and equal angles.
- Apply the rule: For regular polygons, the number of lines of symmetry equals the number of sides. Therefore, it has lines of symmetry.
- Describe locations: lines pass through the opposite vertices (corners), and 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 ways to fold the shape into matching halves.
Problem 2:
A student thinks that a standard non-square rectangle has lines of symmetry because they counted the diagonals. Is the student correct? Explain why or why not.
Solution:
- Analyze the horizontal and vertical folds: Folding a rectangle vertically or horizontally through the center results in matching halves ( lines).
- 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.
- Conclusion: The student is incorrect. A rectangle has only 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.