Review the key concepts, formulae, and examples before starting your quiz.
🔑Concepts
A figure is said to have line symmetry if it can be folded along a line so that the two halves match exactly. Imagine a butterfly: if you draw a vertical line through its center, the left wing is a mirror image of the right wing, appearing as two identical parts facing each other.
The 'Line of Symmetry' or 'Axis of Symmetry' is the line that divides a figure into two identical parts. This line can be vertical (like the letter ), horizontal (like the letter ), or diagonal. For instance, a square has four such lines: one vertical, one horizontal, and two diagonal lines passing through its corners.
Reflection symmetry is closely related to mirror reflections. When an object is reflected in a mirror, the image is the same size but has its left and right sides reversed. The line of symmetry acts like a mirror; the distance from the line to a point on the object is the same as the distance from the line to the corresponding point on the reflected image.
To complete a symmetric figure on a grid or squared paper, use the squares to measure distances precisely. If a vertex of the shape is units to the left of a vertical line of symmetry, you must plot its corresponding vertex units to the right of the line at the same horizontal level.
Regular polygons, which have all sides and angles equal, possess multiple lines of symmetry. The number of lines of symmetry in a regular polygon is equal to the number of its sides. For example, a regular pentagon (5 sides) has lines of symmetry, each passing from a vertex to the midpoint of the opposite side.
Symmetry can be created using the 'Ink-blot' method. By folding a piece of paper, placing a drop of ink on the fold, and pressing the halves together, you create a symmetric pattern where the fold represents the axis of symmetry. The resulting shape is perfectly balanced on both sides.
In geometric constructions, we use the property that the line of symmetry is the perpendicular bisector of the line segment joining a point and its symmetric image. This means the line of symmetry meets the segment at and divides it into two equal lengths.
📐Formulae
💡Examples
Problem 1:
Given a vertical line of symmetry and a point located to the left of , determine the position of its symmetric image and the total distance between and .
Solution:
Step 1: Understand that the line of symmetry acts as a mirror. Step 2: The image will be at the same perpendicular distance from but on the opposite side. Therefore, is to the right of . Step 3: To find the total distance , add the distances from the line: .
Explanation:
In reflection symmetry, the distance from the object to the mirror line is equal to the distance from the image to the mirror line ().
Problem 2:
Identify the number of lines of symmetry in a regular hexagon and describe where they are located.
Solution:
Step 1: Identify that a regular hexagon has sides. Step 2: Use the rule that for a regular polygon, lines of symmetry = . So, there are lines. Step 3: Describe the locations: lines pass through the opposite vertices, and lines pass through the midpoints of the opposite sides.
Explanation:
Regular polygons have a high degree of symmetry. Each line of symmetry divides the hexagon into two congruent (identical) trapezoids or pentagons depending on which axis is used.