11.3 Ratio, Proportion, Scale Drawings, and Geometric Similarity

11.3 Ratio, Proportion, Scale Drawings, and Geometric Similarity

Proportional reasoning and geometric similarity represent Competency 2.2 of the FTCE Mathematics subtest. This domain bridges pure algebra and geometric analysis, evaluating how scale factors transform linear lengths, two-dimensional surface areas, and three-dimensional volumes. Candidates must master both the algebraic procedures of proportion resolution and the conceptual laws governing similar figures.

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Ratios, Rates, and Algebraic Proportions

A ratio expresses a quantitative relationship comparing two numbers or magnitudes through division, written as $a : b$, $a \text{ to } b$, or as a fraction $\frac{a}{b}$ (where $b \ne 0$). Ratios can describe part-to-part relationships (e.g., 12 male students to 16 female students) or part-to-whole relationships (e.g., 12 male students to 28 total students).

A proportion is an equation stating that two ratios are strictly equal:

$$\frac{a}{b} = \frac{c}{d}$$

The Cross-Multiplication Property

In any valid proportion, the product of the extremes ($a$ and $d$) equals the product of the means ($b$ and $c$):

$$\frac{a}{b} = \frac{c}{d} \iff a \cdot d = b \cdot c$$

Worked Example: Multi-Term Algebraic Proportion

Solve for $x$ in the algebraic proportion:

$$\frac{3x - 2}{4} = \frac{2x + 7}{5}$$

• Step 1: Apply cross-multiplication: $$5(3x - 2) = 4(2x + 7)$$
• Step 2: Distribute through the parentheses: $$15x - 10 = 8x + 28$$
• Step 3: Isolate the variable term by subtracting $8x$ from both sides: $$7x - 10 = 28$$
• Step 4: Add 10 to both sides and divide by 7: $$7x = 38 \implies x = \frac{38}{7} \approx 5.43$$

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Geometric Similarity: Definition and Criteria

Two geometric figures are similar (denoted by the symbol $\sim$) if and only if they satisfy two rigid conditions:

1. All pairs of corresponding interior angles are strictly congruent (equal in measure).
2. All pairs of corresponding side lengths are strictly proportional (sharing an identical constant ratio).

                               GEOMETRIC SIMILARITY
                    Preimage (Original)              Image (Dilated)
                           B                              E
                          /│                             / │
                         / │                            /  │
                     c  /  │  a                    kc  /   │  ka
                       /   │                          /    │
                      /____│                         /_____│
                     A   b   C                      D   kb   F
                      ΔABC ~ ΔDEF                     Scale Factor = k

Triangle Similarity Theorems

To establish that two triangles are similar, it is not necessary to verify every side and angle. Three geometric theorems provide shortcuts:

1. Angle-Angle (AA) Similarity: If two angles of one triangle are congruent to two angles of another triangle, the two triangles are similar (since the third angles must sum to 180° and are therefore also congruent).
2. Side-Side-Side (SSS) Similarity: If all three pairs of corresponding sides of two triangles are proportional, the triangles are similar: $$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k$$
3. Side-Angle-Side (SAS) Similarity: If two pairs of corresponding sides are proportional and their included angles are congruent, the triangles are similar.

Indirect Measurement: Shadow Reckoning

Shadow problems appear frequently on the FTCE exam. Because the sun's rays strike the earth at effectively identical angles at any given moment and location, a vertical object and its horizontal ground shadow form a right triangle similar to that formed by any nearby vertical object and its shadow by AA similarity:

$$\frac{\text{Height}_1}{\text{Shadow}_1} = \frac{\text{Height}_2}{\text{Shadow}_2}$$

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Scale Drawings, Blueprints, and Cartographic Scales

A scale drawing or blueprint is a proportionally reduced or enlarged two-dimensional representation of an object. The scale defines the constant conversion ratio between the drawing measurement and the actual real-world measurement:

$$\text{Scale} = \frac{\text{Drawing Measurement}}{\text{Actual Measurement}}$$

When working with blueprints, scales are frequently given with mixed units (such as $\frac{1}{4}\text{ in} = 1\text{ ft}$). To convert safely without errors:

• Method A (Direct Ratio): Set up a proportion using the blueprint unit in the numerator and real-world feet in the denominator.
• Method B (Unitless Scale Factor): Convert both terms into identical units. For example, since $1\text{ ft} = 12\text{ in}$, a scale of $\frac{1}{4}\text{ in} = 1\text{ ft}$ becomes $\frac{0.25\text{ in}}{12\text{ in}} = \frac{1}{48}$. The real object is 48 times larger in every linear dimension.

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The Dimensional Scaling Law: Linear, Area, and Volume Transformations

One of the most frequently missed concepts on teacher certification examinations is how scale factors impact different geometric dimensions. If every linear dimension of an object is enlarged or reduced by a constant scale factor $k$:

                               THE DIMENSIONAL SCALING LAW
                 ┌────────────────────────────────────────────────────────┐
                 │   Linear Dimensions (Length, Width, Height, Radius):   │
                 │                   Scale by factor k¹                   │
                 └───────────────────────────┬────────────────────────────┘
                                             ▼
                 ┌────────────────────────────────────────────────────────┐
                 │   Perimeter & Circumference: Scale by factor k¹        │
                 │   Surface Area & Base Area:  Scale by factor k²        │
                 │   Internal Space & Volume:   Scale by factor k³        │
                 └────────────────────────────────────────────────────────┘

Mathematical Proof of Dimensional Scaling

Consider an original rectangular prism with dimensions $l$, $w$, and $h$:

• Original Area of Base: $A = lw$
• Original Volume: $V = lwh$

If all three dimensions are scaled by factor $k$, the new dimensions become $l' = kl$, $w' = kw$, and $h' = kh$:

• New Base Area: $A' = (kl)(kw) = k^2(lw) = k^2 A$
• New Volume: $V' = (kl)(kw)(kh) = k^3(lwh) = k^3 V$

Exam Trap Alert (Blueprint Area): If a floor plan uses the scale $1\text{ in} = 5\text{ ft}$, a room measuring $3\text{ in} \times 4\text{ in}$ on paper has an area of $12\text{ in}^2$. The real floor area is NOT $12 \times 5 = 60\text{ ft}^2$! Because area scales by $k^2$, the real area is $12 \times 5^2 = 12 \times 25 = 300\text{ ft}^2$. Always convert linear dimensions first ($3 \times 5 = 15\text{ ft}$ and $4 \times 5 = 20\text{ ft}$; $15 \times 20 = 300\text{ ft}^2$) to avoid this trap.