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Vectors and Transformations - Transformations (Reflection, Rotation, Translation, Enlargement)

Grade 10IGCSE

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

🔑Concepts

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Object and Image: The original shape is the 'Object', and the transformed shape is the 'Image' (usually denoted with prime symbols, e.g., A').

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Translation: Moving a shape by a fixed distance in a specified direction using a column vector.

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Reflection: Flipping a shape over a mirror line; every point and its image are equidistant from the line.

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Rotation: Turning a shape around a fixed 'Center of Rotation' by a specific angle and direction (clockwise or anticlockwise).

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Enlargement: Changing the size of a shape from a 'Center of Enlargement' by a 'Scale Factor' (k). If |k| < 1, the shape shrinks.

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Congruency and Similarity: Translation, Reflection, and Rotation produce congruent images. Enlargement produces similar images.

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Invariant Points: Points that do not change position after a transformation (e.g., points on the mirror line in a reflection).

📐Formulae

Translation Vector: T=(xy)T = \begin{pmatrix} x \\ y \end{pmatrix}, where xx is horizontal shift and yy is vertical shift.

Reflection in yy-axis: (x,y)→(−x,y)(x, y) \rightarrow (-x, y)

Reflection in xx-axis: (x,y)→(x,−y)(x, y) \rightarrow (x, -y)

Reflection in y=xy = x: (x,y)→(y,x)(x, y) \rightarrow (y, x)

Rotation 90∘90^{\circ} Clockwise about origin: (x,y)→(y,−x)(x, y) \rightarrow (y, -x)

Rotation 90∘90^{\circ} Anticlockwise about origin: (x,y)→(−y,x)(x, y) \rightarrow (-y, x)

Rotation 180∘180^{\circ} about origin: (x,y)→(−x,−y)(x, y) \rightarrow (-x, -y)

Enlargement Scale Factor: k=Image LengthObject Lengthk = \frac{\text{Image Length}}{\text{Object Length}}

Area of Enlarged Image: Areaimage=k2×Areaobject\text{Area}_{\text{image}} = k^2 \times \text{Area}_{\text{object}}

💡Examples

Problem 1:

Triangle A has vertices (1, 2), (3, 2), and (3, 5). Find the coordinates of the image after a translation by the vector (−24)\begin{pmatrix} -2 \\ 4 \end{pmatrix}.

Solution:

(1-2, 2+4) = (-1, 6); (3-2, 2+4) = (1, 6); (3-2, 5+4) = (1, 9).

Explanation:

Add the top value of the vector to the x-coordinates and the bottom value to the y-coordinates of each vertex.

Problem 2:

Reflect the point P(4, -1) in the line y=−xy = -x.

Solution:

P' (1, -4)

Explanation:

To reflect in y=−xy = -x, swap the x and y coordinates and change both their signs: (x,y)→(−y,−x)(x, y) \rightarrow (-y, -x).

Problem 3:

A square with area 5cm25 cm^2 undergoes an enlargement with a scale factor of 3. What is the area of the image?

Solution:

5×32=5×9=45cm25 \times 3^2 = 5 \times 9 = 45 cm^2.

Explanation:

The area of the image is the object's area multiplied by the square of the scale factor (k2k^2).

Problem 4:

Rotate point Q(2, 5) 90∘90^{\circ} anticlockwise about the origin.

Solution:

Q' (-5, 2)

Explanation:

Using the rotation rule for 90∘90^{\circ} anticlockwise (x,y)→(−y,x)(x, y) \rightarrow (-y, x), the x-coordinate becomes the negative of the original y, and the y-coordinate becomes the original x.