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

Grade 9IGCSE

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

🔑Concepts

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A translation moves an object from one position to another without changing its shape, size, or orientation. It is defined by a column vector (xy)\begin{pmatrix} x \\ y \end{pmatrix}, where xx represents the horizontal shift and yy represents the vertical shift.

A triangle translated by the vector (2, 2) on a coordinate plane.
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A reflection flips an object over a line of reflection. Every point on the image is the same distance from the mirror line as the corresponding point on the original object. Common mirror lines include the xx-axis (y=0y=0), the yy-axis (x=0x=0), and the lines y=xy=x and y=−xy=-x.

Reflection of a triangle across the line y=x.
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A rotation turns an object about a fixed point called the center of rotation. To fully describe a rotation, you must specify the angle (e.g., 90∘90^\circ or 180∘180^\circ), the direction (clockwise or anticlockwise), and the center of rotation (e.g., origin (0,0)(0,0)).

A rectangle rotated 90 degrees anticlockwise about the origin.
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Enlargement changes the size of an object by a scale factor kk from a center of enlargement. If ∣k∣>1|k| > 1, the image is larger; if ∣k∣<1|k| < 1, the image is smaller. If kk is negative, the image is inverted on the opposite side of the center.

Triangle enlarged by a scale factor of 2 from the origin.

📐Formulae

Vector Addition: (ab)+(cd)=(a+cb+d)\begin{pmatrix} a \\ b \end{pmatrix} + \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a+c \\ b+d \end{pmatrix}

Scalar Multiplication: k(xy)=(kxky)k \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} kx \\ ky \end{pmatrix}

Magnitude of a vector v⃗=(xy)\vec{v} = \begin{pmatrix} x \\ y \end{pmatrix}: ∣v⃗∣=x2+y2|\vec{v}| = \sqrt{x^2 + y^2}

Enlargement Area Scale Factor: Area of Image=k2×Area of Object\text{Area of Image} = k^2 \times \text{Area of Object}

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

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

💡Examples

Problem 1:

Translate the point A(2,−3)A(2, -3) by the vector v⃗=(−45)\vec{v} = \begin{pmatrix} -4 \\ 5 \end{pmatrix}. Find the coordinates of the image A′A'.

Solution:

A′=(2+(−4),−3+5)=(−2,2)A' = (2 + (-4), -3 + 5) = (-2, 2)

Explanation:

To translate a point, add the xx-component of the vector to the xx-coordinate and the yy-component of the vector to the yy-coordinate.

Problem 2:

A triangle with vertices P(1,1)P(1, 1), Q(3,1)Q(3, 1), and R(1,4)R(1, 4) is reflected in the line y=xy = x. What are the new coordinates?

Solution:

P′(1,1),Q′(1,3),R′(4,1)P'(1, 1), Q'(1, 3), R'(4, 1)

Explanation:

When reflecting in the line y=xy=x, the xx and yy coordinates of each point are swapped.

Problem 3:

A square is enlarged by a scale factor of 33 from the center (0,0)(0,0). If the original area is 5 cm25 \text{ cm}^2, what is the area of the enlarged square?

Solution:

New Area=5×32=5×9=45 cm2\text{New Area} = 5 \times 3^2 = 5 \times 9 = 45 \text{ cm}^2

Explanation:

The area of an enlarged shape increases by the square of the scale factor (k2k^2).

Problem 4:

Rotate the point (2,5)(2, 5) 90∘90^\circ anticlockwise about the origin (0,0)(0,0).

Solution:

(−5,2)(-5, 2)

Explanation:

For a 90∘90^\circ anticlockwise rotation about the origin, the mapping is (x,y)→(−y,x)(x, y) \rightarrow (-y, x).

Problem 5:

A triangle TT has vertices at A(1,1)A(1, 1), B(2,1)B(2, 1), and C(1,2)C(1, 2). It is enlarged by a scale factor of k=−2k = -2 with the origin (0,0)(0, 0) as the center of enlargement. Determine the new coordinates of the vertices.

Negative enlargement of a triangle through the origin.

Solution:

The rule for enlargement from the origin is (x,y)→(kx,ky)(x, y) \rightarrow (kx, ky). A′(1×−2,1×−2)=(−2,−2)A'(1 \times -2, 1 \times -2) = (-2, -2) B′(2×−2,1×−2)=(−4,−2)B'(2 \times -2, 1 \times -2) = (-4, -2) C′(1×−2,2×−2)=(−2,−4)C'(1 \times -2, 2 \times -2) = (-2, -4)

Explanation:

Since the scale factor is negative, the image is twice as large as the original, but it is inverted and located on the opposite side of the center of enlargement (the origin).

Problem 6:

Reflect the rectangle with vertices R1(2,−1)R_1(2, -1), R2(4,−1)R_2(4, -1), R3(4,−2)R_3(4, -2), and R4(2,−2)R_4(2, -2) in the line x=1x = 1. Find the coordinates of the reflected image.

A rectangle reflected across the vertical line x=1.

Solution:

The mirror line is the vertical line x=1x = 1. The distance from each xx-coordinate to x=1x=1 is mirrored on the other side: R1:x=2R_1: x=2 is 11 unit right of x=1x=1, so R1′R_1' is 11 unit left at x=1−1=0x = 1 - 1 = 0. Coordinates: (0,−1)(0, -1). R2:x=4R_2: x=4 is 33 units right of x=1x=1, so R2′R_2' is 33 units left at x=1−3=−2x = 1 - 3 = -2. Coordinates: (−2,−1)(-2, -1). R3:x=4R_3: x=4 is 33 units right of x=1x=1, so R3′R_3' is 33 units left at x=1−3=−2x = 1 - 3 = -2. Coordinates: (−2,−2)(-2, -2). R4:x=2R_4: x=2 is 11 unit right of x=1x=1, so R4′R_4' is 11 unit left at x=1−1=0x = 1 - 1 = 0. Coordinates: (0,−2)(0, -2).

Explanation:

In a reflection across a vertical line x=ax = a, the yy-coordinates remain the same, and the xx-coordinate changes from xx to 2a−x2a - x. Here, x′=2(1)−xx' = 2(1) - x.