12.2 Coordinate Geometry: Distance, Midpoint, & Geometric Transformations
Key Takeaways
- The 2D Cartesian coordinate plane is structured by perpendicular x- and y-axes intersecting at the origin (0, 0), establishing four quadrants with predictable sign pairs: Q I (+, +), Q II (-, +), Q III (-, -), and Q IV (+, -).
- The Distance Formula d = √((x₂ - x₁)² + (y₂ - y₁)²) derives directly from the Pythagorean Theorem (d² = Δx² + Δy²), while the Midpoint Formula M = ((x₁ + x₂)/2, (y₁ + y₂)/2) calculates the arithmetic center point of a line segment.
- Rigid transformations (translations, reflections, and rotations) preserve side lengths and angle measures (isometry), producing congruent images (≅), whereas non-rigid dilations scale coordinates by factor k ((x, y) → (kx, ky)), producing similar images (∼) where perimeter scales by k and area scales by k².
- Standard coordinate transformation mapping rules include: reflections across the x-axis (x, -y), y-axis (-x, y), and line y = x (y, x); rotations about origin for 90° clockwise (y, -x), 90° counterclockwise (-y, x), and 180° (-x, -y).
Last updated: August 2026
12.2 Coordinate Geometry: Distance, Midpoint, & Geometric Transformations
Coordinate geometry merges algebraic techniques with spatial reasoning. On the TABE 13&14 Mathematics assessment (Levels D and A), examinees must plot and interpret points on the Cartesian coordinate plane, calculate Euclidean distance and midpoint coordinates, solve for missing endpoints, and predict the coordinates of figures under rigid and non-rigid geometric transformations.
The 2D Cartesian Coordinate Plane
The Cartesian coordinate plane is formed by two infinite perpendicular number lines intersecting at the origin $(0, 0)$:
- Horizontal $x$-axis: Positive values extend right ($x > 0$), negative values extend left ($x < 0$).
- Vertical $y$-axis: Positive values extend upward ($y > 0$), negative values extend downward ($y < 0$).
Cartesian Coordinate Plane Quadrants & Signs:
y-axis (Vertical)
▲
|
Quadrant II | Quadrant I
(-, +) | (+, +)
|
-------------------(0,0)-------------------> x-axis (Horizontal)
|
Quadrant III | Quadrant IV
(-, -) | (+, -)
|
▼
| Region / Axis | Sign Pattern | Coordinates Condition | Example Point |
|---|---|---|---|
| Quadrant I (Q I) | $(+, +)$ | $x > 0, \ y > 0$ | $(4, 7)$ |
| Quadrant II (Q II) | $(-, +)$ | $x < 0, \ y > 0$ | $(-5, 3)$ |
| Quadrant III (Q III) | $(-, -)$ | $x < 0, \ y < 0$ | $(-8, -2)$ |
| Quadrant IV (Q IV) | $(+, -)$ | $x > 0, \ y < 0$ | $(6, -4)$ |
| $x$-Axis ($x$-Intercept) | $(x, 0)$ | $y = 0$ (lies on horizontal line) | $(5, 0)$ |
| $y$-Axis ($y$-Intercept) | $(0, y)$ | $x = 0$ (lies on vertical line) | $(0, -3)$ |
| Origin | $(0, 0)$ | Point of intersection | $(0, 0)$ |
The Distance Formula
The Distance Formula computes the straight-line Euclidean distance ($d$) between any two points $(x_1, y_1)$ and $(x_2, y_2)$. It is an algebraic restatement of the Pythagorean Theorem ($a^2 + b^2 = c^2$), where the horizontal run is $\Delta x = (x_2 - x_1)$ and the vertical rise is $\Delta y = (y_2 - y_1)$:
Step-by-Step Distance Algorithm
- Identify and label coordinates $(x_1, y_1)$ and $(x_2, y_2)$.
- Calculate the horizontal difference: $\Delta x = x_2 - x_1$.
- Calculate the vertical difference: $\Delta y = y_2 - y_1$.
- Square both differences: $(\Delta x)^2$ and $(\Delta y)^2$ (note that squaring always eliminates negative signs).
- Sum the squared differences and take the principal square root: $\sqrt{(\Delta x)^2 + (\Delta y)^2}$.
Worked Example: Find the distance between point $A(-3, 2)$ and point $B(5, -4)$.
d &= \sqrt{(5 - (-3))^2 + (-4 - 2)^2} \\ &= \sqrt{(5 + 3)^2 + (-6)^2} \\ &= \sqrt{8^2 + (-6)^2} \\ &= \sqrt{64 + 36} = \sqrt{100} = \mathbf{10\text{ units}} \end{aligned}$$ --- ## The Midpoint Formula & Missing Endpoint Problems The **Midpoint ($M$)** is the exact center point of a line segment connecting $(x_1, y_1)$ and $(x_2, y_2)$. It is found by computing the arithmetic average of the $x$-coordinates and the $y$-coordinates: $$\mathbf{M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)}$$ ### Standard Midpoint Example Find the midpoint of segment $CD$ with endpoints $C(-6, 8)$ and $D(4, -2)$: $$M = \left(\frac{-6 + 4}{2}, \frac{8 + (-2)}{2}\right) = \left(\frac{-2}{2}, \frac{6}{2}\right) = \mathbf{(-1, 3)}$$ ### The "Missing Endpoint" Problem on TABE A frequent TABE Level A question gives the **midpoint $M(x_m, y_m)$** and **one endpoint $(x_1, y_1)$**, requiring you to solve for the coordinates of the other endpoint $(x_2, y_2)$: $$x_2 = 2x_m - x_1 \qquad | \qquad y_2 = 2y_m - y_1$$ **Worked Example:** Line segment $PQ$ has midpoint $M(3, 5)$ and endpoint $P(-1, 2)$. What are the coordinates of endpoint $Q(x_2, y_2)$? - For the $x$-coordinate: $\frac{-1 + x_2}{2} = 3 \implies -1 + x_2 = 6 \implies x_2 = 6 + 1 = \mathbf{7}$ - For the $y$-coordinate: $\frac{2 + y_2}{2} = 5 \implies 2 + y_2 = 10 \implies y_2 = 10 - 2 = \mathbf{8}$ - Endpoint $Q$ is located at $\mathbf{(7, 8)}$. --- ## Geometric Transformations on the Coordinate Plane A **transformation** maps an initial geometric figure (the **pre-image**, denoted $A, B, C$) to a new position (the **image**, denoted $A', B', C'$). ``` Classification of Transformations: ├── Rigid Motions (Isometries - Preserves Size & Shape → Congruence ≅): │ ├── Translation: Sliding horizontally and/or vertically │ ├── Reflection: Flipping across a line of symmetry │ └── Rotation: Turning around a fixed center point (origin) └── Non-Rigid Transformations (Preserves Shape Only → Similarity ∼): └── Dilation: Enlarging or reducing by a scale factor k ``` ### 1. Translations (Rigid Shifts) A translation slides every vertex of a figure by $h$ units horizontally and $k$ units vertically: $$\mathbf{(x, y) \longrightarrow (x + h, \ y + k)}$$ - If $h > 0$, shift right; if $h < 0$, shift left. - If $k > 0$, shift up; if $k < 0$, shift down. ### 2. Reflections (Rigid Flips) A reflection creates a mirror image across a specific line of reflection: - **Reflect across $x$-axis:** Negate the $y$-coordinate: $\mathbf{(x, y) \to (x, -y)}$ - **Reflect across $y$-axis:** Negate the $x$-coordinate: $\mathbf{(x, y) \to (-x, y)}$ - **Reflect across the line $y = x$:** Swap coordinates: $\mathbf{(x, y) \to (y, x)}$ - **Reflect across the line $y = -x$:** Swap and negate coordinates: $\mathbf{(x, y) \to (-y, -x)}$ ### 3. Rotations (Rigid Turns About the Origin $(0, 0)$) Rotations turn a figure around the origin by a specified angle. Unless stated otherwise, mathematical rotations are counterclockwise: - **$90^\circ$ Counterclockwise ($270^\circ$ Clockwise):** $\mathbf{(x, y) \to (-y, x)}$ - **$90^\circ$ Clockwise ($270^\circ$ Counterclockwise):** $\mathbf{(x, y) \to (y, -x)}$ - **$180^\circ$ Rotation (either direction):** $\mathbf{(x, y) \to (-x, -y)}$ ### 4. Dilations (Non-Rigid Scaling) A dilation multiplies both coordinates by a **scale factor ($k$)** with respect to the origin $(0, 0)$: $$\mathbf{(x, y) \longrightarrow (kx, \ ky)}$$ - **Enlargement:** $|k| > 1$ (the image expands away from the origin). - **Reduction:** $0 < |k| < 1$ (the image shrinks toward the origin). ### Dilation Scaling Rules for Length, Perimeter, & Area Under a dilation with scale factor $k$: 1. **Side Lengths:** Scaled by a factor of $\mathbf{k}$ ($s' = k \cdot s$). 2. **Perimeter:** Scaled by a factor of $\mathbf{k}$ ($P' = k \cdot P$). 3. **Area:** Scaled by a factor of $\mathbf{k^2}$ ($A' = k^2 \cdot A$). 4. **Angle Measures:** Remain completely **unchanged** (angle preservation ensures geometric similarity $\sim$). --- ## Multi-Step Transformation Example **Problem:** Triangle $\triangle ABC$ has vertex $A(-2, 4)$. Point $A$ is rotated $90^\circ$ counterclockwise about the origin, and the resulting point $A'$ is translated $5$ units right and $3$ units down to produce $A''$. What are the coordinates of $A''$? 1. Apply $90^\circ$ counterclockwise rotation rule $(x, y) \to (-y, x)$: $$A(-2, 4) \longrightarrow A'(-4, -2)$$ 2. Apply translation rule $(x, y) \to (x + 5, y - 3)$: $$A'(-4, -2) \longrightarrow A''(-4 + 5, -2 - 3) = \mathbf{A''(1, -5)}$$Loading diagram...
Test Your Knowledge
Line segment AB on a Cartesian coordinate grid has one endpoint at A(-5, 7) and a midpoint at M(1, 2). What are the coordinates of the other endpoint, B?
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Test Your Knowledge
A point with coordinates P(-3, 5) is rotated 90° clockwise about the origin, and then translated 4 units right and 2 units down. What are the final coordinates of the image point P''?
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Test Your Knowledge
A rectangular blueprint region on a coordinate plane has an original area of 24 square centimeters. A graphic designer applies a dilation centered at the origin with a scale factor of k = 3.5. What is the area of the dilated image?
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