12.3 Coordinate Geometry, Graphing & Geometric Transformations

Coordinate Geometry, Graphing & Geometric Transformations

Quick Answer: On the WEST-B Mathematics subtest (Objective 0015), coordinate geometry tests your ability to find distances ($d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$), calculate midpoints ($M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$), and execute transformations. Key transformation rules include: Reflections across the $x$-axis ($(x,-y)$), $y$-axis ($(-x,y)$), and line $y=x$ ($(y,x)$); Rotations (CCW about origin) of $90^\circ$ ($(-y, x)$), $180^\circ$ ($(-x, -y)$), and $270^\circ$ ($(y, -x)$); and Dilations centered at the origin ($(kx, ky)$), which multiply perimeter by $k$ and area by $k^2$.

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1. The Cartesian Coordinate Plane & Basic Formulas

The Cartesian coordinate plane is constructed from two perpendicular real number lines intersecting at the origin $(0,0)$.

                                  y-axis
                                    ^
                                    |
                    Quadrant II     |    Quadrant I
                     (- , +)        |     (+ , +)
                                    |
             -----------------------+-----------------------> x-axis
                                    |      Origin (0,0)
                    Quadrant III    |    Quadrant IV
                     (- , -)        |     (+ , -)
                                    |
                                    v

Core Coordinate Geometry Formulas

+---------------------------------------------------------------------------------------------------+
|                             COORDINATE PLANE FORMULAS REFERENCE                                   |
|                                                                                                   |
|   +-----------------------+----------------------------------+--------------------------------+   |
|   | FORMULA NAME          | ALGEBRAIC EXPRESSION             | GEOMETRIC DERIVATION / USE     |   |
|   +-----------------------+----------------------------------+--------------------------------+   |
|   | Midpoint Formula      | M = ((x1 + x2)/2, (y1 + y2)/2)   | Averages the x and y coords    |   |
|   +-----------------------+----------------------------------+--------------------------------+   |
|   | Distance Formula      | d = √((x2 - x1)² + (y2 - y1)²)   | Derived directly from          |   |
|   |                       |                                  | Pythagorean Theorem (Δx² + Δy²)|   |
|   +-----------------------+----------------------------------+--------------------------------+   |
|   | Slope Formula (m)     | m = (y2 - y1) / (x2 - x1)        | Vertical rise over horiz. run  |   |
|   +-----------------------+----------------------------------+--------------------------------+   |
|   | Parallel Slope Rule   | m1 = m2                          | Equal slopes, distinct y-int   |   |
|   +-----------------------+----------------------------------+--------------------------------+   |
|   | Perpendicular Slope   | m1 × m2 = -1 (m2 = -1/m1)        | Opposite reciprocals           |   |
|   +-----------------------+----------------------------------+--------------------------------+   |
+---------------------------------------------------------------------------------------------------+

Finding a Missing Endpoint Given Midpoint

If midpoint $M(x_m, y_m)$ and one endpoint $A(x_1, y_1)$ are known, the missing endpoint $B(x_2, y_2)$ is calculated by:

$$x_2 = 2x_m - x_1, \qquad y_2 = 2y_m - y_1$$

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2. Geometric Transformations

Transformations map a pre-image shape to an image shape in the plane. They are categorized into rigid isometries (preserving size and shape) and non-rigid transformations (changing size).

+---------------------------------------------------------------------------------------------------+
|                                TRANSFORMATION TAXONOMY MATRIX                                     |
|                                                                                                   |
|   +-------------------+---------------+--------------------+----------------------------------+   |
|   | TRANSFORMATION    | TYPE          | PRESERVES          | ALGEBRAIC RULE                   |   |
|   +-------------------+---------------+--------------------+----------------------------------+   |
|   | Translation       | Rigid Motion  | Distance, Angle,   | (x, y) → (x + h, y + k)          |   |
|   | (Slide)           | (Isometry)    | Orientation        |                                  |   |
|   +-------------------+---------------+--------------------+----------------------------------+   |
|   | Reflection        | Rigid Motion  | Distance, Angle;   | Across x-axis: (x, -y)           |   |
|   | (Flip)            | (Isometry)    | Reverses orient.   | Across y-axis: (-x, y)           |   |
|   |                   |               |                    | Across y = x: (y, x)             |   |
|   +-------------------+---------------+--------------------+----------------------------------+   |
|   | Rotation          | Rigid Motion  | Distance, Angle,   | 90° CCW: (-y, x)                 |   |
|   | (Turn)            | (Isometry)    | Orientation        | 180°: (-x, -y)                   |   |
|   |                   |               |                    | 270° CCW: (y, -x)                |   |
|   +-------------------+---------------+--------------------+----------------------------------+   |
|   | Dilation          | Non-Rigid     | Angle measures     | Centered at origin:              |   |
|   | (Scale / Resize)  | (Similarity)  | (produces ~ shape) | (x, y) → (kx, ky)                |   |
|   +-------------------+---------------+--------------------+----------------------------------+   |
+---------------------------------------------------------------------------------------------------+

Detailed Coordinate Rules for Transformations

A. Reflections

• Across the $x$-axis ($y = 0$): $(x, y) \to (x, -y)$ (Sign of $y$ flips)
• Across the $y$-axis ($x = 0$): $(x, y) \to (-x, y)$ (Sign of $x$ flips)
• Across the line $y = x$: $(x, y) \to (y, x)$ (Coordinates swap places)
• Across the line $y = -x$: $(x, y) \to (-y, -x)$ (Coordinates swap and both negate)
• Across the origin $(0,0)$: $(x, y) \to (-x, -y)$ (Equivalent to $180^\circ$ rotation)

B. Rotations (Centered at the Origin)

Unless stated otherwise, mathematical rotations are counterclockwise (CCW):

• $90^\circ$ Counterclockwise (or $270^\circ$ Clockwise): $(x, y) \to (-y, x)$
• $180^\circ$ (Clockwise or Counterclockwise): $(x, y) \to (-x, -y)$
• $270^\circ$ Counterclockwise (or $90^\circ$ Clockwise): $(x, y) \to (y, -x)$
• $360^\circ$ Full Rotation: $(x, y) \to (x, y)$

C. Dilations (Centered at the Origin)

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

• If $k > 1$, the transformation is an enlargement.
• If $0 < k < 1$, the transformation is a reduction.
• Impact on Measurements:
  • Side lengths are multiplied by $k$.
  • Perimeter is multiplied by $k$.
  • Area is multiplied by $k^2$.
  • Angle measures remain unchanged (similarity).

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3. Reflectional & Rotational Symmetry

+---------------------------------------------------------------------------------------------------+
|                                 SYMMETRY CONCEPTS & DEFINITIONS                                   |
|                                                                                                   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | SYMMETRY TYPE         | DEFINITION                  | EXAMPLES & PROPERTIES               |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Line (Reflectional)   | Figure can be folded along  | Regular n-gon has n lines of symm;  |   |
|   | Symmetry              | a line so both halves match | Rectangle has 2; Square has 4;      |   |
|   |                       | exactly                     | Equilateral triangle has 3          |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Rotational Symmetry   | Figure maps onto itself by  | Order n: matches n times in 360°;   |   |
|   |                       | a rotation of ≤ 180°        | Fundamental Angle = 360° / order    |   |
|   |                       | (or strictly < 360°)        | Square: Order 4, Angle = 90°        |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Point Symmetry        | Figure maps onto itself by  | Same as 180° rotational symmetry;   |   |
|   |                       | 180° rotation about center  | Parallelograms, Rectangles, Circles |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
+---------------------------------------------------------------------------------------------------+

Common Shape Symmetry Breakdown

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4. Step-by-Step Worked Problems & Derivations

Problem 1: Finding an Unknown Endpoint

Problem: Line segment $\overline{AB}$ has endpoint $A(-5, 8)$ and midpoint $M(1, 3)$. Find the coordinates of endpoint $B(x_2, y_2)$.

Step-by-Step Solution:

1. State the midpoint coordinates: $$x_m = 1, \quad y_m = 3, \quad x_1 = -5, \quad y_1 = 8$$
2. Apply the inverse midpoint formula for $x_2$: $$\frac{x_1 + x_2}{2} = x_m \implies \frac{-5 + x_2}{2} = 1$$ $$-5 + x_2 = 2 \implies x_2 = 2 + 5 = 7$$
3. Apply the inverse midpoint formula for $y_2$: $$\frac{y_1 + y_2}{2} = y_m \implies \frac{8 + y_2}{2} = 3$$ $$8 + y_2 = 6 \implies y_2 = 6 - 8 = -2$$
4. Conclusion: Endpoint $B = (7, -2)$. Verify: $\left(\frac{-5 + 7}{2}, \frac{8 + (-2)}{2}\right) = \left(\frac{2}{2}, \frac{6}{2}\right) = (1, 3)$. Correct!

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Problem 2: Dilation and Area Transformation

Problem: A right triangle has vertices $P(2, 4)$, $Q(6, 4)$, and $R(2, 1)$. The triangle is dilated by a scale factor of $k = 3$ centered at the origin to form $\triangle P'Q'R'$. What is the area of the dilated image $\triangle P'Q'R'$?

Step-by-Step Solution:

1. Calculate side lengths and area of original $\triangle PQR$:
   • Base leg $\overline{PQ}$ runs horizontally from $x=2$ to $x=6$ at $y=4$: $\text{base} = 6 - 2 = 4\text{ units}$.
   • Height leg $\overline{PR}$ runs vertically from $y=1$ to $y=4$ at $x=2$: $\text{height} = 4 - 1 = 3\text{ units}$.
   • Original Area $= \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 3 = 6\text{ units}^2$.
2. Apply the Area Dilation Factor ($k^2$): $$\text{Area}(\triangle P'Q'R') = k^2 \times \text{Area}(\triangle PQR) = 3^2 \times 6 = 9 \times 6 = 54\text{ units}^2$$
3. Alternative verification via coordinates:
   • $P'(6, 12)$, $Q'(18, 12)$, $R'(6, 3)$.
   • Dilated base $= 18 - 6 = 12\text{ units}$.
   • Dilated height $= 12 - 3 = 9\text{ units}$.
   • Dilated Area $= \frac{1}{2} \times 12 \times 9 = 54\text{ units}^2$. Both methods confirm $54$.

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Problem 3: Composite Coordinate Transformation

Problem: Point $T(-3, 5)$ is rotated $90^\circ$ counterclockwise about the origin, and its image is then reflected across the $x$-axis. Determine the final coordinates of the point.

Step-by-Step Solution:

1. Step 1: Rotation of $90^\circ$ Counterclockwise:
   • Rule: $(x, y) \to (-y, x)$
   • Apply to $T(-3, 5)$: $x = -3, y = 5$
   • $T' = (-5, -3)$
2. Step 2: Reflection across the $x$-axis:
   • Rule: $(x, y) \to (x, -y)$
   • Apply to $T'(-5, -3)$: $x = -5, y = -3$
   • $T'' = (-5, -(-3)) = (-5, 3)$
3. Conclusion: The final coordinates are $(-5, 3)$.