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Shape and Space - Transformations (Translation, Reflection, Rotation)

Grade 6IB

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

🔑Concepts

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Introduction to Transformations: A transformation is a process that moves or changes a geometric figure (called the 'pre-image') to create a new figure (called the 'image'). In Grade 6, we focus on rigid transformations—Translation, Reflection, and Rotation—where the shape and size remain identical, meaning the image is congruent to the pre-image. Visually, if you were to cut out the pre-image, it would fit perfectly over the image.

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Translation (The Slide): A translation moves every point of a shape the same distance in the same direction. It is often described by a translation vector (xy)\binom{x}{y}, where the top number represents horizontal movement (right is positive, left is negative) and the bottom number represents vertical movement (up is positive, down is negative). Visually, the shape 'slides' across the coordinate plane without turning or resizing.

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Reflection (The Flip): A reflection flips a figure over a specific line called the 'axis of reflection' or mirror line. Every point on the image is at the exact same distance from the mirror line as the corresponding point on the pre-image, but on the opposite side. If you visualize folding the grid along the mirror line, the pre-image and the image would match up exactly.

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Rotation (The Turn): A rotation turns a figure around a fixed point known as the 'center of rotation.' To describe a rotation fully, you must provide the angle (such as 90∘90^{\circ} or 180∘180^{\circ}), the direction (clockwise or counter-clockwise), and the coordinates of the center. Visually, it looks like a wheel spinning; points closer to the center move in smaller arcs, while points further away move in larger arcs.

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The Center of Rotation and Mirror Lines: For Grade 6, the center of rotation is most commonly the origin (0,0)(0, 0). Common mirror lines for reflection include the xx-axis (the horizontal line where y=0y=0), the yy-axis (the vertical line where x=0x=0), and the diagonal line y=xy = x. Identifying these lines visually is key to performing transformations correctly.

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Prime Notation: To distinguish between the original shape and its new position, we use 'prime' notation. If the original vertex is labeled AA, the new position after a transformation is labeled A′A'. If the shape is moved again, it becomes A′′A'' (double-prime). This helps us track the movement of specific corners of a polygon.

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Invariance: During translations, reflections, and rotations, certain properties of the shape remain 'invariant' (unchanged). These include the lengths of the sides, the measures of the interior angles, and the total area. The only things that change are the coordinates and the orientation (the direction the shape is facing).

📐Formulae

Translation by vector (ab):(x,y)→(x+a,y+b)\binom{a}{b}: (x, y) \rightarrow (x + a, y + b)

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

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

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

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

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

💡Examples

Problem 1:

A triangle has vertices at A(1,2)A(1, 2), B(4,2)B(4, 2), and C(1,5)C(1, 5). Translate the triangle using the vector (−32)\binom{-3}{2} and state the new coordinates of vertex A′A'.

Solution:

  1. Identify the translation values: a=−3a = -3 (move 3 units left) and b=2b = 2 (move 2 units up).
  2. Apply the formula (x+a,y+b)(x + a, y + b) to the original coordinates of A(1,2)A(1, 2).
  3. Calculate the new x-coordinate: x′=1+(−3)=−2x' = 1 + (-3) = -2.
  4. Calculate the new y-coordinate: y′=2+2=4y' = 2 + 2 = 4.
  5. The new coordinates for vertex A′A' are (−2,4)(-2, 4).

Explanation:

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

Problem 2:

Point PP is located at (3,−5)(3, -5). Reflect point PP across the yy-axis and then describe the position of the resulting image P′P'.

Solution:

  1. Identify the reflection rule for the yy-axis: (x,y)→(−x,y)(x, y) \rightarrow (-x, y).
  2. The original xx coordinate is 33, so the new xx coordinate is −(3)=−3-(3) = -3.
  3. The yy coordinate remains unchanged: y=−5y = -5.
  4. The image P′P' is located at (−3,−5)(-3, -5).

Explanation:

When reflecting across the yy-axis, the point moves horizontally to the opposite side of the vertical axis, so only the sign of the xx-coordinate changes.