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Geometry and Measurement - Transformational Geometry (Translation, Reflection, Rotation)

Grade 8IB

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

🔑Concepts

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Transformations and Notation: A transformation is a process that maps an original geometric figure, called the pre-image, onto a new figure called the image. The image is typically labeled using prime notation (e.g., if the pre-image is point AA, the image is A′A'). Visually, this creates a relationship between two shapes on a coordinate plane where every point on the first shape has a corresponding point on the second.

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Translation (Sliding): A translation moves every point of a figure the same distance in a specific direction without changing its size, shape, or orientation. Visually, the object 'slides' across the grid. It is often described by a translation vector (ab)\begin{pmatrix} a \\ b \\ \end{pmatrix}, where aa represents the horizontal shift (right is positive, left is negative) and bb represents the vertical shift (up is positive, down is negative).

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Reflection (Flipping): A reflection creates a mirror image of a figure across a specific line called the line of reflection. Each point in the image is the same distance from the line of reflection as the corresponding point in the pre-image, but on the opposite side. Visually, if you folded the graph paper along the line of reflection, the two shapes would overlap perfectly.

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Rotation (Turning): A rotation turns a figure about a fixed point called the center of rotation by a specific angle and direction (clockwise or counter-clockwise). In Grade 8, the center of rotation is most commonly the origin (0,0)(0, 0). Visually, the shape 'swings' around the center like a hand on a clock, changing its orientation but not its size or shape.

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Isometry and Congruence: Translations, reflections, and rotations are called rigid transformations or 'isometries.' Because these movements do not stretch or shrink the figure, the pre-image and the image are always congruent. This means they have the exact same side lengths, angle measures, and area.

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Lines of Symmetry: A line of symmetry is a line that divides a figure into two mirror-image halves. Some shapes have multiple lines of symmetry (like a square with 4), while others have none. Visually, this concept is closely tied to reflection, as a shape with line symmetry is its own image when reflected across that line.

📐Formulae

Translation rule: (x,y)→(x+a,y+b)(x, y) \to (x + a, y + b)

Reflection over the x-axis: (x,y)→(x,−y)(x, y) \to (x, -y)

Reflection over the y-axis: (x,y)→(−x,y)(x, y) \to (-x, y)

Reflection over the line y=xy = x: (x,y)→(y,x)(x, y) \to (y, x)

Reflection over the line y=−xy = -x: (x,y)→(−y,−x)(x, y) \to (-y, -x)

Rotation 90∘90^\circ clockwise about origin: (x,y)→(y,−x)(x, y) \to (y, -x)

Rotation 90∘90^\circ counter-clockwise about origin: (x,y)→(−y,x)(x, y) \to (-y, x)

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

💡Examples

Problem 1:

Triangle ABCABC has vertices A(1,2)A(1, 2), B(4,2)B(4, 2), and C(1,5)C(1, 5). Apply a translation using the vector (−31)\begin{pmatrix} -3 \\ 1 \\ \end{pmatrix} and then reflect the resulting image over the x-axis. Find the final coordinates of vertex C′′C''.

Solution:

Step 1: Apply the translation to point C(1,5)C(1, 5). Using the rule (x+a,y+b)(x + a, y + b), we get C′(1+(−3),5+1)=C′(−2,6)C'(1 + (-3), 5 + 1) = C'(-2, 6).\Step 2: Apply the reflection over the x-axis to C′(−2,6)C'(-2, 6). Using the rule (x,−y)(x, -y), we get C′′(−2,−6)C''(-2, -6).\The final coordinates of C′′C'' are (−2,−6)(-2, -6).

Explanation:

To find the final position, we apply the transformations sequentially. The translation shifts the point 3 units left and 1 unit up. The reflection over the x-axis then negates the y-coordinate while keeping the x-coordinate the same.

Problem 2:

A square has a vertex at S(2,−3)S(2, -3). If the square is rotated 90∘90^\circ counter-clockwise about the origin, what are the coordinates of the image S′S'?

Solution:

Step 1: Identify the starting coordinates (x,y)=(2,−3)(x, y) = (2, -3).\Step 2: Apply the rotation rule for 90∘90^\circ counter-clockwise, which is (x,y)→(−y,x)(x, y) \to (-y, x).\Step 3: Substitute the values: x′=−(−3)=3x' = -(-3) = 3 and y′=2y' = 2.\The coordinate of S′S' is (3,2)(3, 2).

Explanation:

A 90∘90^\circ counter-clockwise rotation swaps the x and y values and changes the sign of the original y-coordinate. Visually, the point moves from Quadrant IV to Quadrant I.