krit.club logo

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

Grade 11A Level

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

🔑Concepts

•

A Translation moves every point of a shape the same distance in a specified direction, defined by a column vector (ab)\begin{pmatrix} a \\ b \end{pmatrix}, where aa is the horizontal shift and bb is the vertical shift.

Diagram showing triangle A translated to A' by vector (3, 3).
•

Reflection creates a mirror image of a shape across a line of reflection. Every point and its image are equidistant from this line.

Reflection of a triangle across the y-axis.
•

Rotation turns a shape about a fixed point called the center of rotation. It is defined by the center, the angle, and the direction (clockwise or anti-clockwise).

Rotation of a triangle 90 degrees anti-clockwise about the origin.
•

Enlargement changes the size of a shape by a scale factor kk from a center of enlargement. If k>1k > 1 the shape grows; if 0<k<10 < k < 1 it shrinks; if k<0k < 0 it is inverted on the opposite side of the center.

Enlargement of a triangle from center (0,0) with scale factor 2.

📐Formulae

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

Problem 5:

Reflect the rectangle with vertices P(1,1)P(1, 1), Q(3,1)Q(3, 1), R(3,2)R(3, 2), and S(1,2)S(1, 2) in the line y=−1y = -1. State the coordinates of the image vertices.

Rectangle reflected across the line y = -1.

Solution:

  1. Identify the mirror line y=−1y = -1.
  2. Calculate the vertical distance of each point from y=−1y = -1. Point P(1,1)P(1, 1) is 2 units above the line (1−(−1)=21 - (-1) = 2).
  3. Place the image point P′P' the same distance below the line: y′=−1−2=−3y' = -1 - 2 = -3. Thus P′(1,−3)P'(1, -3).
  4. Repeating for others: Q′(3,−3)Q'(3, -3), R′(3,−4)R'(3, -4), S′(1,−4)S'(1, -4).

Explanation:

In a reflection across a horizontal line y=cy = c, the xx-coordinate remains unchanged, while the yy-coordinate changes such that the line y=cy = c is the midpoint between the original yy and the image y′y'.

Problem 6:

Rotate the shape with vertices (1,1)(1, 1), (2,1)(2, 1), and (1,3)(1, 3) 90∘90^\circ clockwise about the point (1,1)(1, 1).

Triangle rotated 90 degrees clockwise about the vertex (1,1).

Solution:

  1. The center of rotation is (1,1)(1, 1).
  2. Point A(1,1)A(1, 1) is the center, so its image A′A' remains (1,1)(1, 1).
  3. Point B(2,1)B(2, 1) is 1 unit to the right of the center. Rotating 90∘90^\circ clockwise moves it 1 unit down from the center: (1,0)(1, 0).
  4. Point C(1,3)C(1, 3) is 2 units above the center. Rotating 90∘90^\circ clockwise moves it 2 units to the right of the center: (3,1)(3, 1). Image vertices: (1,1),(1,0),(3,1)(1, 1), (1, 0), (3, 1).

Explanation:

When rotating about a point (h,k)(h, k) other than the origin, subtract the center coordinates, apply the rotation rule, then add the center coordinates back.