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Geometry - Symmetry and transformations (reflection/rotation)

Grade 6IGCSE

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

🔑Concepts

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Line Symmetry: A shape has line symmetry if it can be divided into two identical halves by a line (mirror line), where one side is a mirror image of the other.

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Rotational Symmetry: The number of times a shape looks exactly the same as its original position while being rotated through a full 360∘360^\circ turn. This is called the 'Order of Symmetry'.

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Reflection: A transformation that 'flips' a figure over a line. Every point on the image is the same distance from the mirror line as the corresponding point on the original object.

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Rotation: A transformation that 'turns' a figure around a fixed point called the Center of Rotation. It is defined by the center, the angle of rotation (e.g., 90∘,180∘90^\circ, 180^\circ), and the direction (clockwise or anti-clockwise).

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Congruence: In both reflection and rotation, the original shape and the image are congruent, meaning they have the same size and shape.

📐Formulae

Order of Rotational Symmetry (Regular Polygon) = n, where nn is the number of sides.

Reflection Property: Distance(Object, Mirror Line) = Distance(Image, Mirror Line).

Full Turn = 360∘360^\circ; Half Turn = 180∘180^\circ; Quarter Turn = 90∘90^\circ.

💡Examples

Problem 1:

A square is rotated around its center. What is its order of rotational symmetry?

Solution:

Order 4

Explanation:

A square looks identical to its starting position at rotations of 90∘90^\circ, 180∘180^\circ, 270∘270^\circ, and 360∘360^\circ. Since it fits onto itself 4 times in a full circle, the order is 4.

Problem 2:

Reflect the point A(3,5)A(3, 5) in the x-axis. What are the coordinates of the image A′A'?

Solution:

A′(3,−5)A'(3, -5)

Explanation:

When reflecting in the x-axis, the x-coordinate remains the same, but the y-coordinate changes its sign (positive becomes negative) because it moves to the opposite side of the horizontal axis.

Problem 3:

Identify the number of lines of symmetry in a regular pentagon.

Solution:

5

Explanation:

A regular pentagon has 5 equal sides and 5 equal angles. You can draw a line of symmetry from each vertex to the midpoint of the opposite side, resulting in 5 lines.

Problem 4:

Rotate the point (2,0)(2, 0) by 90∘90^\circ anti-clockwise about the origin (0,0)(0, 0).

Solution:

(0,2)(0, 2)

Explanation:

An anti-clockwise rotation of 90∘90^\circ moves a point on the positive x-axis to the positive y-axis. The distance from the origin remains 2 units.