14.1 Coordinate Geometry, Linear Graphs, and Geometric Transformations

14.1 Coordinate Geometry, Linear Graphs, and Geometric Transformations

CSET Focus: The California Subject Examinations for Teachers (CSET) Multiple Subjects Subtest II requires candidates to demonstrate mastery of coordinate geometry and geometric transformations. You must be prepared to calculate distances, midpoints, and slopes on the Cartesian plane; convert between linear equation forms; determine perpendicular and parallel relationships; apply algebraic mapping rules for rigid motions (translations, reflections, rotations) and dilations; and explain the mathematical principles underlying line symmetry, rotational symmetry, and Euclidean tessellations.

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1. The Cartesian Coordinate Plane & Analytical Line Formulas

The Cartesian coordinate plane is formed by the perpendicular intersection of two real number lines at the origin $(0,0)$. The horizontal axis is designated as the $x$-axis (abscissa) and the vertical axis as the $y$-axis (ordinate). These axes partition the infinite two-dimensional plane into four distinct quadrants numbered counterclockwise:

                          y-axis
                            │
        Quadrant II         │        Quadrant I
        (-x, +y)            │        (+x, +y)
                            │
   ─────────────────────────┼───────────────────────── x-axis
                            │ (0,0) Origin
        Quadrant III        │        Quadrant IV
        (-x, -y)            │        (+x, -y)
                            │

Analytical Coordinate Formulas

1. The Distance Formula

The Euclidean distance $d$ between two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ is directly derived from the Pythagorean Theorem ($a^2 + b^2 = c^2$) by constructing a right triangle with legs parallel to the coordinate axes:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

• Example: Find the distance between $A(-3, 5)$ and $B(3, -3)$:

$$d = \sqrt{(3 - (-3))^2 + (-3 - 5)^2} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$$

2. The Midpoint Formula

The midpoint $M$ of the line segment connecting $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ represents the coordinate-wise arithmetic mean of the endpoints:

$$M = \left( \frac{x_1 + x_2}{2}, \; \frac{y_1 + y_2}{2} \right)$$

• Example: The midpoint of segment $AB$ connecting $(-4, 7)$ and $(8, 1)$ is:

$$M = \left( \frac{-4 + 8}{2}, \; \frac{7 + 1}{2} \right) = \left( \frac{4}{2}, \; \frac{8}{2} \right) = (2, 4)$$

3. Slope and Rate of Change

The slope ($m$) of a non-vertical line measures its steepness and direction, defined as the ratio of vertical change (rise, $\Delta y$) to horizontal change (run, $\Delta x$):

$$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \quad (x_1 \ne x_2)$$

• Positive Slope ($m > 0$): Line rises from left to right.
• Negative Slope ($m < 0$): Line falls from left to right.
• Zero Slope ($m = 0$): Horizontal line ($y = k$), where $\Delta y = 0$ and $\Delta x \ne 0$.
• Undefined Slope: Vertical line ($x = h$), where $\Delta x = 0$ and division by zero is undefined.

Forms of Linear Equations

1. Slope-Intercept Form: $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept $(0, b)$.
2. Point-Slope Form: $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is any specific point on the line.
3. Standard Form: $Ax + By = C$, where $A, B, C \in \mathbb{Z}$, $A \ge 0$, and $A$ and $B$ are not both zero. The slope of a standard form line is $m = -\frac{A}{B}$ and the $y$-intercept is $(0, \frac{C}{B})$.

Parallel and Perpendicular Line Relationships

• Parallel Lines ($L_1 \parallel L_2$): Two non-vertical lines are parallel if and only if their slopes are equal ($m_1 = m_2$) and their $y$-intercepts are distinct ($b_1 \ne b_2$). Coincident lines share both identical slopes and identical intercepts.
• Perpendicular Lines ($L_1 \perp L_2$): Two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals (opposite reciprocals):

$$m_1 \cdot m_2 = -1 \iff m_2 = -\frac{1}{m_1}$$

• Horizontal and Vertical Special Case: Any horizontal line ($y = k$, $m = 0$) is perpendicular to any vertical line ($x = h$, undefined slope).

Slope Relationships & Linear Forms Matrix

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2. Geometric Transformations: Rigid Motions (Isometries)

A transformation is a bijection (one-to-one and onto mapping) that assigns each pre-image point $P(x, y)$ in the plane to a unique image point $P'(x', y')$. An isometry (or rigid motion) is a transformation that strictly preserves Euclidean distance:

$$d(P, Q) = d(P', Q') \quad \text{for all points } P, Q$$

Rigid motions also preserve angle measure, collinearity, betweenness, and area. The image figure is congruent to the pre-image figure ($\Delta ABC \cong \Delta A'B'C'$).

                         GEOMETRIC TRANSFORMATIONS
                   ┌──────────────────┴──────────────────┐
             RIGID MOTIONS (Isometries)           NON-RIGID (Similarity)
         (Preserve distance, angle, area)        (Preserves angle, shape)
         ┌───────────┼───────────┐                         │
    TRANSLATIONS  REFLECTIONS  ROTATIONS               DILATIONS
     (Slide: v)   (Flip: Line) (Turn: Center θ)        (Scale: k)
     [Direct]      [Opposite]     [Direct]             [Direct]

1. Translations (Slides)

A translation shifts every point in the plane by a constant horizontal distance $a$ and vertical distance $b$, represented by the translation vector $\vec{v} = \langle a, b \rangle$:

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

Translations are direct isometries because they preserve the geometric orientation (handedness) of vertices.

2. Reflections (Flips)

A reflection flips a figure over a specified line of reflection $L$. The line $L$ acts as the perpendicular bisector of the line segment connecting each pre-image point $P$ to its image point $P'$. Reflections are opposite isometries because they reverse the cyclic clockwise/counterclockwise orientation of vertices.

Standard Coordinate Reflection Rules:

• Reflection across the $x$-axis ($y = 0$): $(x, y) \longrightarrow (x, -y)$
• Reflection across the $y$-axis ($x = 0$): $(x, y) \longrightarrow (-x, y)$
• Reflection across the line $y = x$: $(x, y) \longrightarrow (y, x)$
• Reflection across the line $y = -x$: $(x, y) \longrightarrow (-y, -x)$
• Reflection across a vertical line $x = h$: $(x, y) \longrightarrow (2h - x, y)$
• Reflection across a horizontal line $y = k$: $(x, y) \longrightarrow (x, 2k - y)$

3. Rotations (Turns)

A rotation turns every point around a fixed point called the center of rotation by a specified directed angle $\theta$. In standard coordinate geometry, positive angles represent counterclockwise (CCW) rotations about the origin $(0,0)$. Rotations are direct isometries.

Standard Origin-Centered Rotation Rules:

• $90^\circ$ CCW Rotation (or $270^\circ$ CW): $(x, y) \longrightarrow (-y, x)$
• $180^\circ$ Rotation (Half-Turn): $(x, y) \longrightarrow (-x, -y)$
• $270^\circ$ CCW Rotation (or $90^\circ$ CW): $(x, y) \longrightarrow (y, -x)$
• $360^\circ$ Rotation (Full Turn / Identity): $(x, y) \longrightarrow (x, y)$

Coordinate Transformation Mapping Rules Table

CSET Diagnostic Trap: When performing compositions of transformations, order matters! A reflection followed by a translation generally does not yield the same resulting coordinates as the translation followed by the reflection. Always evaluate transformations step-by-step in the sequence specified.

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3. Non-Rigid Transformations: Dilations & Similarity

A dilation is a non-rigid transformation that expands or compresses a geometric figure relative to a fixed center of dilation $C$ by a positive scalar factor $k$ ($k > 0$).

Properties of Dilations

1. Origin-Centered Coordinate Mapping:

$$(x, y) \longrightarrow (kx, ky)$$

1. Scale Factor Regimes:
   • Enlargement ($k > 1$): The image is strictly larger than the pre-image ($d(C, P') = k \cdot d(C, P)$).
   • Reduction ($0 < k < 1$): The image is strictly smaller than the pre-image.
   • Identity ($k = 1$): The image is congruent to the pre-image.
2. Preservation of Invariants: Dilations preserve angle measure, collinearity, betweenness, and parallelism of lines. Consequently, dilations produce similar figures ($\Delta ABC \sim \Delta A'B'C'$).
3. Dimensional Scaling Ratios:
   • Linear Dimensions (Side Lengths, Perimeter, Altitude): Scaled by factor $k$. $\text{Perimeter}' = k \times \text{Perimeter}$
   • Two-Dimensional Quantities (Surface Area, Base Area): Scaled by factor $k^2$. $\text{Area}' = k^2 \times \text{Area}$
   • Three-Dimensional Quantities (Volume, Capacity): Scaled by factor $k^3$. $\text{Volume}' = k^3 \times \text{Volume}$

• Worked Example: A triangle with an area of $24\text{ cm}^2$ is dilated by a scale factor of $k = 3$. The perimeter of the new triangle is $3$ times the original, but its area becomes $3^2 \times 24 = 9 \times 24 = 216\text{ cm}^2$.

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4. Symmetry and Euclidean Tessellations

Symmetry Classifications

• Line Symmetry (Reflectional / Bilateral): A figure possesses line symmetry if there exists a line $L$ such that reflecting the figure over $L$ maps the figure entirely onto itself. A regular $n$-sided polygon has exactly $n$ lines of symmetry.
• Rotational Symmetry: A figure possesses rotational symmetry if it maps onto itself after a rotation of angle $\theta$ where $0^\circ < \theta < 360^\circ$ about its geometric center.
  • Order of Rotational Symmetry ($n$): The number of distinct rotational positions in $[0^\circ, 360^\circ)$ in which the figure looks identical.
  • Angle of Rotational Symmetry: The minimum positive angle of rotation: $\theta = \frac{360^\circ}{n}$
  • Example: A regular hexagon has rotational symmetry of order $6$, with a fundamental angle of $\frac{360^\circ}{6} = 60^\circ$ (mapping onto itself at $60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ, 360^\circ$).
• Point Symmetry: Equivalent to rotational symmetry of order 2 ($180^\circ$ rotation), where every point $(x, y)$ corresponds to a matching point $(-x, -y)$ equidistant from the center.

Tessellations (Tilings) of the Plane

A tessellation (or tiling) is a pattern covering an infinite plane with repeating geometric shapes without any gaps or overlaps.

The Vertex Condition for Regular Tessellations

A regular tessellation uses only one type of regular polygon (monohedral tiling). For regular polygons to tile the plane, the sum of the interior angles of the polygons meeting at every shared vertex must equal exactly $360^\circ$.

The interior angle $\theta$ of a regular $n$-sided polygon is given by:

$$\theta = \frac{(n - 2) \times 180^\circ}{n}$$

                                  Euclidean Tessellation Condition
                                 k × θ_interior = 360° (k ∈ ℤ⁺)
          ┌───────────────────────────────────┼───────────────────────────────────┐
  Equilateral Triangle                      Square                          Regular Hexagon
  n = 3, θ = 60°                            n = 4, θ = 90°                  n = 6, θ = 120°
  6 × 60° = 360°                            4 × 90° = 360°                  3 × 120° = 360°
  [6 triangles at vertex]                   [4 squares at vertex]           [3 hexagons at vertex]

The Three Regular Tessellations in Euclidean Geometry

1. Equilateral Triangles ($n = 3$): Interior angle $= 60^\circ$. Since $6 \times 60^\circ = 360^\circ$, exactly $6$ triangles meet at each vertex (vertex configuration $3.3.3.3.3.3$).
2. Squares ($n = 4$): Interior angle $= 90^\circ$. Since $4 \times 90^\circ = 360^\circ$, exactly $4$ squares meet at each vertex (vertex configuration $4.4.4.4$).
3. Regular Hexagons ($n = 6$): Interior angle $= 120^\circ$. Since $3 \times 120^\circ = 360^\circ$, exactly $3$ hexagons meet at each vertex (vertex configuration $6.6.6$).

Why Regular Pentagons Cannot Tessellate: For a regular pentagon ($n = 5$), the interior angle is $\frac{(5-2) \times 180^\circ}{5} = 108^\circ$. Three pentagons sum to $3 \times 108^\circ = 324^\circ$ (leaving a $36^\circ$ gap), while four pentagons sum to $432^\circ$ (overlapping). Since $108^\circ$ does not divide $360^\circ$ evenly, regular pentagons cannot form a regular tessellation.