8.3 Dilations, Scale Factors & Proportional Reasoning in Geometry

8.3 Dilations, Scale Factors & Proportional Reasoning in Geometry

1. Foundations of Dilations: Centers, Scale Factors & Invariants

A dilation is a non-rigid transformation that alters a figure's size while preserving its shape. A dilation $D_{C, k}$ is defined by a center of dilation $C$ and a non-zero scale factor $k$. For any point $P$, its image $P'$ satisfies:

1. Point $P'$ lies on the line passing through $C$ and $P$ ($\overleftrightarrow{CP}$).
2. Distance from $C$ scales by the absolute factor: $CP' = |k| \cdot CP$.

Scale Factor Taxonomy

• Enlargement ($|k| > 1$): Image is larger than pre-image.
• Reduction ($0 < |k| < 1$): Image is smaller than pre-image.
• Identity ($k = 1$): Every point maps to itself.
• Positive Factor ($k > 0$): Image $P'$ lies on the same ray $\overrightarrow{CP}$ as $P$. Orientation is preserved.
• Negative Factor ($k < 0$): Image $P'$ lies on the opposite ray through $C$. A negative dilation equals a positive dilation of scale factor $|k|$ composed with a $180^\circ$ rotation about $C$: $D_{C, k} = R_{C, 180^\circ} \circ D_{C, |k|}$ When $k = -1$, the transformation is a point reflection (or $180^\circ$ rotation) through $C$.

Invariants of Dilations

Dilations preserve:

• Angle Measure: Angles are congruent (dilations are conformal).
• Parallelism: If $L_1 \parallel L_2$, then $L_1' \parallel L_2'$. If line $L$ does not contain center $C$, image $L'$ is a distinct line parallel to $L$ ($L' \parallel L$). If $L$ contains $C$, line $L$ maps onto itself ($L' = L$).
• Collinearity and Betweenness: Points on a common line remain collinear in the same order.
• Orientation: Polygons retain clockwise or counterclockwise vertex ordering.

Dilations do not preserve distance: segment lengths scale by $|k|$.

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2. Coordinate Formulations of Dilations

In the Cartesian coordinate plane, dilations are evaluated using coordinate mapping rules.

Dilation Centered at the Origin

When centered at origin $O(0, 0)$, coordinates multiply directly by $k$: $D_{O, k}(x, y) = (kx, ky)$

Dilation Centered at an Arbitrary Point $C(x_0, y_0)$

When centered at an arbitrary point $C(x_0, y_0)$, the transformation applies a three-step vector process:

1. Translate center $C$ to the origin: $(x - x_0, y - y_0)$.
2. Scale relative coordinates by $k$: $(k(x - x_0), k(y - y_0))$.
3. Translate back by adding $(x_0, y_0)$:

$D_{C(x_0, y_0), k}(x, y) = \big(x_0 + k(x - x_0), \; y_0 + k(y - y_0)\big)$

Worked Exemplar: Dilation About an Arbitrary Center

Dilate point $P(7, -1)$ with scale factor $k = -2$ centered at $C(3, 2)$.

1. Relative displacement vector: $\vec{v} = (x - x_0, y - y_0) = (7 - 3, \; -1 - 2) = (4, -3)$
2. Scale displacement by $k = -2$: $k \cdot \vec{v} = -2(4, -3) = (-8, 6)$
3. Add center coordinates: $(x', y') = (3 + (-8), \; 2 + 6) = (-5, 8)$
4. Verification: Distance $CP = \sqrt{4^2 + (-3)^2} = 5$. Distance $CP' = \sqrt{(-8)^2 + 6^2} = 10$. The ratio $CP'/CP = 10/5 = 2 = |-2|$, and $\vec{CP'} = -2\vec{CP}$ confirms the $180^\circ$ direction reversal.

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3. Dimensional Scaling Relationships Across Dimensions

When a figure scales by a linear factor $k > 0$, measurements across dimensions scale by powers of $k$:

Structural Justification

Any planar figure partitioned into $N$ triangles of area $\Delta A_i = \frac{1}{2} b_i h_i$ scales to $\Delta A_i' = \frac{1}{2}(kb_i)(kh_i) = k^2 \Delta A_i$. Summing yields $\text{Area}' = k^2 \sum \Delta A_i = k^2 \cdot \text{Area}$. Similarly, summing infinitesimal cuboids of volume $\Delta V_i = \Delta x_i \Delta y_i \Delta z_i$ yields an aggregate volume expansion of $k^3$.

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4. Solving Multi-Dimensional Proportional Reasoning Problems

Secondary mathematics assessments frequently test inverse scaling relationships, transitioning between area and volume ratios.

The Dimensional Bridge Principle

Always compute the underlying linear scale factor $k$ before converting between area and volume:

1. Given area ratio $\frac{A_2}{A_1} = r$, solve for $k = \sqrt{r}$.
2. Given volume ratio $\frac{V_2}{V_1} = s$, solve for $k = \sqrt[3]{s}$.
3. Volume from area ratio: $\frac{V_2}{V_1} = k^3 = (\sqrt{r})^3 = r^{3/2}$.
4. Surface area from volume ratio: $\frac{A_2}{A_1} = k^2 = (\sqrt[3]{s})^2 = s^{2/3}$.

Applied Exemplar: Container Capacity and Material Cost

A storage container of height $0.8$ m holds $25$ L and costs $$12.00$ in metal sheet material for its outer surface. A similar container has height $2.4$ m.

• Linear scale factor: $k = \frac{2.4}{0.8} = 3$.
• Capacity (Volume): Scales by $k^3 = 3^3 = 27$. Capacity $= 25 \times 27 = 675$ L.
• Material Cost (Surface Area): Scales by $k^2 = 3^2 = 9$. Cost $= $12.00 \times 9 = $108.00$.