Unit 10: Ratios, Proportions, and Similar Polygons Notes

Ratios and Proportions

• Ratio Definition: A ratio is an expression that compares two quantities by DIVISION.

• Written Forms: Ratios can be represented in three primary ways:

  • As a fraction: $\frac{a}{b}$ (This form is most commonly used for Probability).

  • Using the word "to": $a \text{ to } b$.

  • Using a colon: $a:b$ (This form is most commonly used for Odds).

Probability and Odds

• Probability: The ratio of successful outcomes compared to the total number of possible outcomes.

  • Formula: $\text{Probability} = \frac{\text{Successful outcomes}}{\text{Total outcomes}}$

• Odds: The ratio of successful outcomes compared to unsuccessful outcomes.

  • Formula: $\text{Odds} = \text{Successful outcomes} : \text{Unsuccessful outcomes}$

• Total Integrity Building: A key relationship in these calculations is that the sum of successes and unsuccesses equals the total outcomes.

  • Formula: $\text{Success} + \text{Unsuccess} = \text{TOTAL outcomes}$

Application Examples

• Marble Jar Scenario: You have 3 blue marbles, 5 red marbles, and 2 green marbles in a jar.

  • Total marbles: $3 + 5 + 2 = 10$.

  • Probability of choosing a red marble: $\frac{5}{10} = \frac{1}{2}$.

  • Odds of choosing a blue marble: Successes (3 blue) : Unsuccesses (5 red + 2 green = 7). Result: $3:7$.

• Grid Analysis Example: Finding the ratio in a grid of 12 total units with 4 units shaded.

  • Ratio of Shaded to Whole Area: $\frac{4}{12}$, which simplifies to $\frac{1}{3}$.

  • Ratio of Shaded to Unshaded Area: $\frac{4}{8}$, which simplifies to $\frac{1}{2}$ or $1:2$.

  • Probability of Shaded Area: This is indicated by the shaded-to-whole ratio ($\frac{1}{3}$).

  • Odds of Shaded Area: This is indicated by the shaded-to-unshaded ratio ($1:2$).

• Conversion Examples:

  • If the probability is $\frac{7}{10}$, then the odds are $7:3$.

  • If the odds are $1:4$, then the probability is $\frac{1}{5}$.

Proportions and Cross-Multiplication

• Proportion Definition: A proportion is a statement of equality between two ratios.

• Equality Property: If $\frac{a}{b} = \frac{c}{d}$, then it can be cross-multiplied to solve for unknowns.

  • Cross-Multiplication Formula: $a \cdot d = b \cdot c$

  • These parts are interchangeable; $\frac{a}{c} = \frac{b}{d}$ would also result in the same cross-product: $a \cdot d = b \cdot c$.

Practical Problem Solving

• Work/Earnings Scenario: If you make $\$180$ every 3 weeks, how much do you make in 8 weeks?

  • Method 1 (Proportion): $\frac{180}{3} = \frac{x}{8}$

    • $3x = 180 \times 8$

    • $3x = 1440$

    • $x = \$480$

  • Method 2 (Unit Rate): Calculate earnings per week first ($\frac{180}{3} = \$60/\text{wk}$). Multiply the unit rate by the target duration ($8 \times 60 = \$480$).

Similar Polygons

• Congruent Figures: Figures that have the EXACT SAME SHAPE and EXACT SAME SIZE. (Symbol: $\cong$)

• Similar Figures: Figures that have the SAME SHAPE, but a DIFFERENT SIZE. These are the result of a reduction or enlargement. (Symbol: $\sim$)

Criteria for Similarity and Congruency

• Congruency (Review): Two polygons are congruent if and only if:

  1. Corresponding angles are congruent ($\cong$).

  2. Corresponding sides are congruent ($\cong$).

  • Example: In $\triangle ABC \cong \triangle DEF$, $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$, and $AB=DE$, $BC=EF$, $AC=DF$.

• Similarity (New): Two polygons are similar if and only if:

  1. Corresponding angles are congruent ($\cong$).

  2. Corresponding sides are proportional.

  • Example: In $\triangle ABC \sim \triangle DEF$, $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$, but side ratios are equal: $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$.

Similarity Proofs and Examples

• Checking Scaling Factor ($ABCD \sim WXYZ$):

  • Verify angles: $\angle A = \angle W$, $\angle B = \angle X$, $\angle C = \angle Y$, $\angle D = \angle Z$.

  • Verify side proportions: $\frac{AB}{WX} = \frac{12}{6}$, $\frac{BC}{XY} = \frac{10}{5}$, $\frac{CD}{YZ} = \frac{8}{4}$, $\frac{AD}{WZ} = \frac{6}{3}$.

  • All ratios simplify to $2$, which is the Scaling Factor.

• Rectangles (Non-similar example):

  • Even though all angles in two rectangles are $90^{\circ}$, they are not similar if sides are not proportional.

  • Example: One rectangle is $4 \times 5$, another is $2 \times 9$.

  • $\frac{4}{2} = 2$, but $\frac{5}{9} \neq 2$. Cross-check: $9 \cdot 2 = 18$ and $4 \cdot 5 = 20$. Since $18 \neq 20$, they are not similar.

• Parallelograms (Non-similar example):

  • Even if sides are proportional ($\frac{12}{6} = 2$ and $\frac{18}{9} = 2$), if the corresponding angles are not equal (e.g., $60^{\circ}$ vs. $42^{\circ}$), the figures are not similar.

Solving for Unknowns in Similar Polygons

• Example: Given $STAR \sim BUDE$:

  • To find angle $y$: Identify the corresponding angle ($\angle S$ corresponds to $\angle B$). $\angle S = 54^{\circ}$, so $y = 54^{\circ}$. Note: It is not $42^{\circ}$.

  • To find side $x$: Use corresponding side ratios. $\frac{SR}{BE} = \frac{TR}{DE}$.

  • Proportion: $\frac{6}{18} = \frac{16}{x}$

  • Cross-multiply: $6x = 18 \cdot 16$

  • $6x = 288$

  • $x = 48$ (Note: Common error might lead to $x=64$, but $48$ is correct).

• Example: Given $\triangle BAD \sim \triangle RAT$ (Twist + Flip):

  • Small triangle side $12$ corresponds to large triangle side $18$.

  • Small triangle side $x$ corresponds to large triangle side $12$.

  • Proportion: $\frac{x}{12} = \frac{12}{18}$

  • Simplified view: $\frac{x}{12} = \frac{3}{4}$ leads to $4x = 36$.

  • Result: $x = 9$.