Comprehensive Guide to Similar Polygons: Properties, Scale Factors, and Theorems

Definition and Properties of Similar Polygons

• Similar polygons are figures that have the same shape but not necessarily the same size.

• The mathematical notation for similarity is a tilde symbol ($\sim$).

• For two polygons to be considered similar, they must satisfy two specific conditions:

  • All corresponding angles must be congruent ($\cong$).

  • All corresponding side lengths must be in proportion.

• Proportion in this context means that the ratios of corresponding side lengths are equal (fractions equal fractions), allowing for the use of cross-multiplication to solve for unknown values.

Identifying Corresponding Parts and Order of Statements

• The order of vertices in a similarity statement is crucial because it identifies the corresponding parts.

• Example: Polygon ABCD is similar to Polygon EFGH ($ABCD \sim EFGH$)

  • Corresponding Angles:

    • The first letter corresponds to the first letter: $Angle \, A = Angle \, E$.

    • The second letter corresponds to the second letter: $Angle \, B = Angle \, F$.

    • The third letter corresponds to the third letter: $Angle \, C = Angle \, G$.

    • The fourth letter corresponds to the fourth letter: $Angle \, D = Angle \, H$.

  • Corresponding Sides:

    • The ratios are formed by taking pairs of letters in order: $\frac{AB}{EF} = \frac{BC}{FG} = \frac{CD}{GH} = \frac{DA}{HE}$.

• Consistency in ratios: When setting up proportions, always keep the same figure in the numerator and the same figure in the denominator (e.g., "the red guy over the blue guy").

Verification of Similarity: Triangles RST and XYZ

• Case Study: $\Delta RST \sim \Delta XYZ$

  • Angle Verification:

    • $\angle R \cong \angle X$

    • $\angle S \cong \angle Y$

    • $\angle T \cong \angle Z$

  • Side Length Ratio Verification:

    • Bottom sides ($RS$ and $XY$): $\frac{20}{12} = \frac{5}{3}$ (Reduced by a factor of 4).

    • Right sides ($ST$ and $YZ$): $\frac{30}{18} = \frac{5}{3}$ (Reduced by a factor of 6).

    • Left sides ($TR$ and $ZX$): $\frac{25}{15} = \frac{5}{3}$ (Reduced by a factor of 5).

  • Since all corresponding sides reduce to the same ratio ($\frac{5}{3}$) and all corresponding angles are equal, the triangles are confirmed similar.

The Scale Factor of Similar Polygons

• The scale factor is defined as the reduced ratio of the lengths of corresponding sides of two similar polygons.

• Example Evaluation:

  • Comparing side lengths of two figures: $\frac{32}{40}, \frac{24}{30}, \frac{16}{20}, \frac{20}{25}$.

  • All these ratios reduce to a scale factor of $\frac{4}{5}$.

  • Because the angles are marked as congruent and the sides share a consistent scale factor of $\frac{4}{5}$, the polygons are similar.

Solving for Missing Values in Similar Figures

• Example 1: $\Delta DEF \sim \Delta MNP$

  • Identifying the scale factor: One set of corresponding sides is given as 9 and 12, so the scale factor is $\frac{9}{12}$.

  • Using the scale factor to find a missing "slanty" side length $x$:

    • $\frac{9}{12} = \frac{x}{\text{corresponding side length}}$

    • By cross-multiplying, the result is determined to be $x = 15$.

• Example 2: $ABCD \sim QRST$

  • Identifying the scale factor using the bottom sides: $\frac{16}{8} = \frac{2}{1}$.

  • Note: It is important to express the scale factor as a ratio ($\frac{2}{1}$) rather than just the integer 2, as this assists in solving proportions.

  • Finding the value of $x$ (where $x$ corresponds to a side of length 4):

    • $\frac{2}{1} = \frac{x}{4}$

    • Cross-multiplying yields $x = 8$.

Theorems for Perimeters and Altitudes

• Ratio of Perimeters Theorem: If two polygons are similar, then the ratio of their perimeters is equal to the scale factor of the polygons.

• Ratio of Altitudes Theorem: The ratio of the lengths of corresponding altitudes of similar triangles is equal to the scale factor of those triangles.

Practical Application: Similar Pentagons

• Example Comparison of Two Similar Pentagons:

  • Scale Factor: A side of length 10 on the first pentagon corresponds to a side of 15 on the second. The scale factor is $\frac{10}{15} = \frac{2}{3}$.

  • Finding a Side Length (x):

    • A left-side length $x$ on the first corresponds to 18 on the second.

    • $\frac{2}{3} = \frac{x}{18}$

    • $3x = 2 \times 18$

    • $3x = 36$

    • $x = 12$

  • Finding Perimeter:

    • The perimeter of the larger pentagon is calculated or given as 69.

    • Let $y$ be the perimeter of the smaller pentagon.

    • $\frac{y}{69} = \frac{2}{3}$

    • $3y = 2 \times 69$

    • $3y = 138$

    • $y = 46$ (The perimeter of the smaller pentagon).

Practical Application: Finding Altitudes

• Example: $\Delta TPR \sim \Delta XPZ$

  • Goal: Find the length of the altitude $PS$.

  • Scale Factor: Determine using sides 12 and 16.

    • $\frac{12}{16} = \frac{3}{4}$.

  • Relationship: $\frac{\text{Scale Factor 1}}{\text{Scale Factor 2}} = \frac{\text{Altitude 1}}{\text{Altitude 2}}$.

  • Setting up the proportion using altitude length 20 on the larger triangle and calling the missing altitude $y$:

    • $\frac{3}{4} = \frac{y}{20}$

    • $4y = 60$

    • $y = 15$

  • Result: The length of the altitude $PS$ is 15.