7.3 Coordinate Plane & Geometric Transformations

7.3 Coordinate Plane & Geometric Transformations

Quick Summary: Transformations are Grades 3-9 curriculum content rather than one of EQAO's published fundamental knowledge and skills for the MPT mathematics component — but the coordinate plane itself is unavoidable, because representing linear relations graphically and solving linear systems graphically are both listed skills, and dilation scale factors are exactly the $k$ versus $k^2$ reasoning that area-scale questions rely on. This section covers rigid transformations (isometries: translations, reflections, rotations) and non-rigid transformations (dilations), coordinate mapping rules, composite multi-step transformations, perimeter and area scaling factors ($k$ vs $k^2$), and line and rotational symmetry.

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Fundamentals of Transformations: Rigid vs. Non-Rigid

A geometric transformation is an operation that moves or alters a geometric figure (the pre-image) to create a new figure (the image).

• Pre-image: The original figure before transformation, labeled with standard vertices like $A, B, C$.
• Image: The new figure after transformation, labeled with prime notation like $A', B', C'$ (or $A'', B'', C''$ after a second transformation).

Classification of Transformations:

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1. Rigid Transformations (Isometries): Maintain side lengths, angle measures, and enclosed area. The pre-image and image are congruent ($\cong$).
   • Translations (slides)
   • Reflections (flips)
   • Rotations (turns)
2. Non-Rigid Transformations: Alter linear dimensions while preserving angle measures and shape proportions. The pre-image and image are similar ($\sim$).
   • Dilations (enlargements and reductions)

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Translations (Slides)

A translation shifts every point of a figure by the same distance in a specified direction along a vector $\langle a, b \rangle$.

Algebraic Coordinate Rule:

$$(x, y) \longrightarrow (x + a, y + b)$$

• If $a > 0$, shift right by $a$ units; if $a < 0$, shift left by $|a|$ units.
• If $b > 0$, shift up by $b$ units; if $b < 0$, shift down by $|b|$ units.

Example: Translating point $P(3, -2)$ by vector shift $\langle -4, 5 \rangle$:

$$P'(3 - 4, -2 + 5) = P'(-1, 3)$$

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Reflections (Flips)

A reflection flips a figure across a specified line of reflection (mirror line). Each point and its reflected image are equidistant from the mirror line, which acts as the perpendicular bisector of the segment connecting $P$ and $P'$.

Standard Algebraic Reflection Rules:

Visual Tip for Line $y=x$: Reflecting across $y=x$ converts $(2, 5)$ into $(5, 2)$. Reflecting across $y=-x$ converts $(2, 5)$ into $(-5, -2)$.

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Rotations (Turns around the Origin)

A rotation turns a figure around a fixed center point (on the MPT, standard rotations are centered at the origin $(0,0)$) through a specified angle of rotation.

Orientation Convention:

• Counterclockwise (CCW): Standard positive direction in mathematics.
• Clockwise (CW): Negative direction. Note that a $90^\circ$ CCW rotation is equivalent to a $270^\circ$ CW rotation.

Standard Origin Rotation Rules:

Example: Rotating point $A(4, 2)$ counterclockwise by $90^\circ$ around the origin yields $A'(-2, 4)$.

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Dilations & Scale Factors (Resizing)

A dilation resizes a figure proportionally relative to a fixed center of dilation (origin $(0,0)$) using a non-zero scale factor $k$.

Algebraic Coordinate Rule:

$$(x, y) \longrightarrow (k \cdot x, k \cdot y)$$

• Enlargement: If $k > 1$, the image is larger than the pre-image.
• Reduction: If $0 < k < 1$, the image is smaller than the pre-image.

Impact of Dilation Scale Factor $k$ on Geometric Properties:

1. Side Lengths: Every linear side length is multiplied by $k$: $s_{\text{image}} = k \cdot s_{\text{pre-image}}$.
2. Perimeter: Total perimeter is multiplied by $k$: $P_{\text{image}} = k \cdot P_{\text{pre-image}}$.
3. Surface Area: Enclosed area is multiplied by $k^2$: $$A_{\text{image}} = k^2 \cdot A_{\text{pre-image}}$$

Critical MPT Rule: If a polygon is enlarged by scale factor $k = 3$, its perimeter triples ($3\times$), but its area increases by $3^2 = 9$ times ($9\times$)!

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Line Symmetry & Rotational Symmetry

1. Line Symmetry (Reflectional Symmetry)

A figure has line symmetry if it can be folded along a line such that the two halves coincide exactly. The folding line is called a line of symmetry.

• Equilateral Triangle: $3$ lines of symmetry.
• Square: $4$ lines of symmetry (2 midsegment lines, 2 diagonal lines).
• Regular Hexagon: $6$ lines of symmetry.
• Circle: Infinite lines of symmetry.

2. Rotational Symmetry

A figure has rotational symmetry if it can be rotated around its center by an angle less than $360^\circ$ and map onto itself.

• Order of Rotation ($n$): The number of times the figure looks identical to its original position during a full $360^\circ$ turn.
• Angle of Rotation ($\theta_{\text{rot}}$): The minimum angle of turn required: $$\theta_{\text{rot}} = \frac{360^\circ}{n}$$

Example: A regular pentagon has order of rotation $n = 5$ and angle of rotation $\theta_{\text{rot}} = \frac{360^\circ}{5} = 72^\circ$.

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Composite Transformations & Worked Examples

A composite transformation is a sequence of two or more transformations applied in order.

Example 1: Multi-Step Rigid Composite Transformation

Problem: Triangle $ABC$ has vertices $A(1, 4)$, $B(3, 1)$, and $C(5, 4)$.

1. Apply Transformation 1: Reflect triangle $ABC$ across the $y$-axis to form triangle $A'B'C'$.
2. Apply Transformation 2: Rotate triangle $A'B'C'$ counterclockwise by $90^\circ$ around the origin to form triangle $A''B''C''$.
3. Determine the final coordinates of vertex $A''$.

Solution:

1. Step 1: Reflect $A(1, 4)$ across the $y$-axis Rule: $(x, y) \longrightarrow (-x, y)$.

   $$A'( -1, 4 )$$

2. Step 2: Rotate $A'(-1, 4)$ CCW by $90^\circ$ around origin Rule: $(x, y) \longrightarrow (-y, x)$. Here $x = -1$ and $y = 4$.

   $$A''(-4, -1)$$

Final Answer: Vertex $A''$ has coordinates $(-4, -1)$.

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Example 2: Dilation Scale Factor & Area Scaling

Problem: A rectangular floor plan on a blueprint has vertices at $(0,0)$, $(8,0)$, $(8,5)$, and $(0,5)$ in centimetres.

1. Calculate the area of the blueprint rectangle in square centimetres.
2. A contractor dilates the plan with a scale factor of $k = 40$ to construct the actual building floor.
3. Calculate the perimeter and area of the actual building floor in metres and square metres.

Solution:

1. Step 1: Blueprint Area

   $$\text{Length} = 8\text{ cm}, \quad \text{Width} = 5\text{ cm}$$ $$\text{Area}_{\text{blueprint}} = 8 \times 5 = 40\text{ cm}^2$$

2. Step 2: Actual Dimensions & Perimeter

   $$\text{Actual Length} = 8\text{ cm} \times 40 = 320\text{ cm} = 3.2\text{ m}$$ $$\text{Actual Width} = 5\text{ cm} \times 40 = 200\text{ cm} = 2.0\text{ m}$$ $$\text{Perimeter}_{\text{actual}} = 2(3.2 + 2.0) = 2(5.2) = 10.4\text{ metres}$$

3. Step 3: Actual Area Calculation Using area scaling factor $k^2 = 40^2 = 1,600$:

   $$\text{Area}_{\text{actual}} = 40\text{ cm}^2 \times 1,600 = 64,000\text{ cm}^2$$

   Converting $\text{cm}^2$ to $\text{m}^2$ (divide by $10,000$):

   $$\text{Area}_{\text{actual}} = \frac{64,000}{10,000} = 6.4\text{ m}^2$$

   Direct Verification: $3.2\text{ m} \times 2.0\text{ m} = 6.4\text{ m}^2$ (Matches!).