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Vectors and Transformations - Transformations: Translation, Reflection, Rotation, Enlargement

Grade 11IGCSE

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

🔑Concepts

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Translation: Moving a shape without rotating or resizing it, defined by a column vector (xy)\begin{pmatrix} x \\ y \end{pmatrix}.

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

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Rotation: Turning a shape around a fixed 'center of rotation' by a specific angle and direction (clockwise or anti-clockwise).

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Enlargement: Changing the size of a shape from a 'center of enlargement' using a scale factor kk.

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Negative Enlargement: The image is on the opposite side of the center and is inverted.

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Invariant Points: Points that do not move under a specific transformation (e.g., points on the mirror line during reflection).

📐Formulae

Translation: (x′y′)=(xy)+(ab)\text{Translation: } \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} a \\ b \end{pmatrix}

Area Scale Factor=k2 (where k is the linear scale factor)\text{Area Scale Factor} = k^2 \text{ (where } k \text{ is the linear scale factor)}

Reflection in y=x:(x,y)→(y,x)\text{Reflection in } y = x: (x, y) \rightarrow (y, x)

Reflection in y=−x:(x,y)→(−y,−x)\text{Reflection in } y = -x: (x, y) \rightarrow (-y, -x)

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

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

💡Examples

Problem 1:

Triangle A has vertices (1, 2), (3, 2), and (1, 5). Find the coordinates of the image after a translation by the vector (−34)\begin{pmatrix} -3 \\ 4 \end{pmatrix}.

Solution:

(1-3, 2+4) = (-2, 6); (3-3, 2+4) = (0, 6); (1-3, 5+4) = (-2, 9).

Explanation:

To translate a point, add the top value of the vector (x-shift) to the x-coordinate and the bottom value (y-shift) to the y-coordinate.

Problem 2:

A square with an area of 5 cm² is enlarged with a scale factor of 3. What is the area of the enlarged square?

Solution:

New Area = 5×32=5×9=45 cm25 \times 3^2 = 5 \times 9 = 45 \text{ cm}^2.

Explanation:

When a shape is enlarged by a scale factor kk, its area increases by k2k^2.

Problem 3:

Describe fully the single transformation that maps Triangle T with vertices (2, 1), (4, 1), (4, 2) onto Triangle U with vertices (-2, -1), (-4, -1), (-4, -2).

Solution:

Rotation of 180∘180^\circ about the origin (0,0)(0,0).

Explanation:

Each point (x,y)(x, y) has been mapped to (−x,−y)(-x, -y). This identifies a 180∘180^\circ rotation. Since the distance from the origin remains proportional and the orientation is flipped both horizontally and vertically, the center is (0,0)(0,0).

Problem 4:

Enlarge triangle ABC with vertices A(1,1), B(2,1), C(1,3) by scale factor -2, center of enlargement (0,0).

Solution:

A'(-2, -2), B'(-4, -2), C'(-2, -6).

Explanation:

Multiply each coordinate by the scale factor -2 because the center is the origin. The negative sign means the image is inverted and on the opposite side of the center.