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Geometry - Symmetry and transformations (reflection/rotation)

Grade 6Cambridge (IGCSE)

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

🔑Concepts

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Line Symmetry (Reflectional Symmetry): A shape has a line of symmetry if it can be folded along a line so that the two halves match exactly. For example, an isosceles triangle has one line of symmetry passing through the vertex between the equal sides.

Isosceles triangle with a vertical dashed line of symmetry.
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Rotational Symmetry: The number of times a shape fits into itself during a full 360∘360^\circ rotation is called its order of rotational symmetry. An equilateral triangle has an order of 3.

Equilateral triangle highlighting the center of rotation.
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Reflections in the Coordinate Plane: When reflecting in the line y=xy = x, the coordinates (x,y)(x, y) map to (y,x)(y, x). Reflecting in the yy-axis changes the sign of the x-coordinate: (x,y)→(−x,y)(x, y) \rightarrow (-x, y).

Point P reflected across the y-axis to P'.
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Rotations on a Grid: A rotation is defined by a center, an angle (90∘90^\circ, 180∘180^\circ, or 270∘270^\circ), and a direction (clockwise or anti-clockwise). Points are moved along circular paths around the center.

📐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.

Problem 5:

Reflect the triangle with vertices A(1,1)A(1, 1), B(4,1)B(4, 1), and C(1,3)C(1, 3) in the line y=−1y = -1. Find the coordinates of the image triangle A′B′C′A'B'C'.

Triangle ABC reflected across the horizontal line y = -1 to form A'B'C'.

Solution:

  1. Identify the distance of each vertex from the mirror line y=−1y = -1.
  2. A(1,1)A(1, 1) is 2 units above y=−1y = -1, so A′A' is 2 units below: A′(1,−3)A'(1, -3).
  3. B(4,1)B(4, 1) is 2 units above y=−1y = -1, so B′B' is 2 units below: B′(4,−3)B'(4, -3).
  4. C(1,3)C(1, 3) is 4 units above y=−1y = -1, so C′C' is 4 units below: C′(1,−5)C'(1, -5).

The coordinates are A′(1,−3)A'(1, -3), B′(4,−3)B'(4, -3), and C′(1,−5)C'(1, -5).

Explanation:

Reflections preserve the distance of every point from the mirror line. Since the line y=−1y = -1 is horizontal, only the y-coordinates change while x-coordinates remain the same.

Problem 6:

A rectangle has vertices at (1,1)(1, 1), (3,1)(3, 1), (3,2)(3, 2), and (1,2)(1, 2). Rotate this rectangle 180∘180^\circ about the origin (0,0)(0, 0).

Rectangle in the first quadrant rotated 180 degrees into the third quadrant.

Solution:

A 180∘180^\circ rotation about the origin maps any point (x,y)(x, y) to (−x,−y)(-x, -y).

  1. (1,1)→(−1,−1)(1, 1) \rightarrow (-1, -1)
  2. (3,1)→(−3,−1)(3, 1) \rightarrow (-3, -1)
  3. (3,2)→(−3,−2)(3, 2) \rightarrow (-3, -2)
  4. (1,2)→(−1,−2)(1, 2) \rightarrow (-1, -2)

The new vertices are (−1,−1)(-1, -1), (−3,−1)(-3, -1), (−3,−2)(-3, -2), and (−1,−2)(-1, -2).

Explanation:

A 180∘180^\circ rotation is equivalent to reflecting in the origin. Both the x and y signs are inverted. The shape remains the same size and orientation is inverted.