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Geometry - Lines of symmetry and reflection

Grade 4Cambridge (IGCSE)

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

🔑Concepts

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A line of symmetry is an imaginary line where you can fold a shape and have both halves match exactly. For a regular polygon, the number of lines of symmetry is equal to the number of sides. For example, a regular triangle has 3 lines of symmetry.

Equilateral triangle showing three lines of symmetry passing through vertices and midpoints.
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In a reflection, every point on the original object and its corresponding point on the image are equidistant from the mirror line. The line connecting a point to its image is perpendicular to the mirror line.

Point A and its image A' shown at equal distances from a vertical mirror line.
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Reflection creates a lateral inversion. This means if you reflect an object in a vertical mirror line, the left and right sides are swapped. The size and shape (congruency) remain the same.

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An object has rotational symmetry if it looks the same after a rotation of less than 360∘360^{\circ} about its center. The number of times it looks the same during a full turn is called the order of rotational symmetry.

📐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}

Distance of Object from Mirror Line=Distance of Image from Mirror Line\text{Distance of Object from Mirror Line} = \text{Distance of Image from Mirror Line}

Orientation of Image=Laterally Inverted (flipped horizontally or vertically)\text{Orientation of Image} = \text{Laterally Inverted (flipped horizontally or vertically)}

💡Examples

Problem 1:

How many lines of symmetry does a non-square rectangle have?

Solution:

2 lines of symmetry.

Explanation:

A rectangle can be folded in half vertically and horizontally to match the sides perfectly. It cannot be folded diagonally because the corners will not meet.

Problem 2:

A point PP is 3 units to the left of a vertical mirror line. Where will its reflected image P′P' be located?

Solution:

3 units to the right of the mirror line.

Explanation:

In reflection, the image is always the same distance from the mirror line as the original object, but on the opposite side.

Problem 3:

Which of these capital letters have at least one line of symmetry: A, F, H, L?

Solution:

A and H.

Explanation:

Letter 'A' has one vertical line of symmetry down the middle. Letter 'H' has two: one vertical and one horizontal. 'F' and 'L' cannot be folded to match perfectly.

Problem 4:

How many lines of symmetry does a regular pentagon have?

Solution:

5 lines of symmetry.

Explanation:

Since a regular pentagon has 5 equal sides and 5 equal angles, it follows the rule that the number of lines of symmetry equals the number of sides. Each line runs from a vertex to the midpoint of the opposite side.

Problem 5:

Reflect the triangle with vertices at A(1,2)A(1, 2), B(3,2)B(3, 2), and C(1,4)C(1, 4) across the vertical line x=4x = 4. What are the coordinates of the reflected image A′B′C′A'B'C'?

Coordinate grid showing a triangle reflected across the vertical line x=4.

Solution:

A′(7,2),B′(5,2),C′(7,4)A'(7, 2), B'(5, 2), C'(7, 4)

Explanation:

To reflect across x=4x = 4, we find the horizontal distance from each point to the line and move the same distance to the other side. A(1,2)A(1,2) is 4−1=34 - 1 = 3 units from the line, so A′A' is at 4+3=74 + 3 = 7. B(3,2)B(3,2) is 4−3=14 - 3 = 1 unit from the line, so B′B' is at 4+1=54 + 1 = 5. C(1,4)C(1,4) is 4−1=34 - 1 = 3 units from the line, so C′C' is at 4+3=74 + 3 = 7. The yy-coordinates remain unchanged.

Problem 6:

Identify the number of lines of symmetry in the following star shape (regular hexagram).

A six-pointed star with several lines of symmetry drawn through its points.

Solution:

66

Explanation:

A regular hexagram (six-pointed star) has 6 lines of symmetry: 3 lines passing through opposite pairs of points, and 3 lines passing through the opposite interior vertices (the 'valleys').