🔷Honors Geometry Unit 7 – Proportions and Similarity

Key Concepts and Definitions

• Ratio expresses the comparison of two quantities with the same units
• Proportion is a statement that two ratios are equal
• Similar figures have the same shape but not necessarily the same size
  • Corresponding angles are congruent
  • Corresponding sides are proportional
• Scale factor is the ratio of corresponding side lengths between two similar figures
• Dilation is a transformation that enlarges or reduces a figure by a scale factor
• Geometric mean of two numbers $a$ and $b$ is the square root of their product $\sqrt{ab}$
• Cross products of a proportion $\frac{a}{b} = \frac{c}{d}$ are equal: $ad = bc$

Ratio and Proportion Basics

• Ratios can be written in fraction form $\frac{a}{b}$, colon form $a:b$, or with the word "to" $a$ to $b$
• Equivalent ratios are created by multiplying or dividing both terms by the same non-zero number
• Proportions are two equivalent ratios, often written as $\frac{a}{b} = \frac{c}{d}$
  • Cross multiplication is used to solve for a missing term: if $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$
• Ratios and proportions can represent part-to-part or part-to-whole relationships
  • Part-to-part example: the ratio of boys to girls in a class is $3:2$
  • Part-to-whole example: the ratio of red marbles to total marbles is $2:5$
• Proportional relationships have a constant ratio between two quantities
  • Example: doubling the length of a rectangle while keeping the width constant maintains the same ratio of length to width

Similar Figures and Shapes

• Similar figures have the same shape but may differ in size
  • Corresponding angles are congruent, and corresponding sides are proportional
• Two polygons are similar if their corresponding angles are congruent and their corresponding side lengths are proportional
• Triangles are similar if they satisfy one of the following conditions:
  • AA (Angle-Angle) similarity: two pairs of corresponding angles are congruent
  • SAS (Side-Angle-Side) similarity: two pairs of corresponding sides are proportional, and the included angle is congruent
  • SSS (Side-Side-Side) similarity: all three pairs of corresponding sides are proportional
• The ratio of perimeters of similar figures is equal to the scale factor
• The ratio of areas of similar figures is equal to the square of the scale factor

Proportional Relationships in Geometry

• In similar triangles, the ratio of corresponding side lengths is constant, forming a proportion
• The geometric mean of two numbers $a$ and $b$ is the length of the side adjacent to both $a$ and $b$ in a right triangle
  • Example: in a right triangle with hypotenuse $13$ and one leg $5$, the other leg is the geometric mean $\frac{5 \cdot 13}{5} = 12$
• The altitude to the hypotenuse of a right triangle creates two similar triangles
  • The altitude is the geometric mean of the two segments of the hypotenuse
• In a right triangle, the length of the altitude to the hypotenuse is the geometric mean of the lengths of the two segments of the hypotenuse
• The Pythagorean theorem relates the side lengths of a right triangle: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse

Similarity Theorems and Proofs

• Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, it divides those sides proportionally
  • Converse: If a line divides two sides of a triangle proportionally, it is parallel to the third side
• Angle Bisector Theorem: The angle bisector of an angle of a triangle divides the opposite side into segments proportional to the lengths of the other two sides
• Triangle Angle Bisector Theorem: The angle bisector of an angle of a triangle divides the opposite side into segments that are proportional to the lengths of the other two sides
• Two triangles are similar if they have two pairs of congruent angles (AA similarity)
  • Proof: If two angles of one triangle are congruent to two angles of another triangle, the third angles must also be congruent (sum of angles in a triangle is 180°), making the triangles similar
• Two triangles are similar if they have two pairs of proportional sides and a congruent included angle (SAS similarity)
  • Proof: If two pairs of sides are proportional and the included angle is congruent, the triangles can be scaled to match the side lengths, resulting in congruent triangles and thus similar triangles

Applications in Real-World Scenarios

• Shadows: The ratio of an object's height to its shadow length is equal to the ratio of another object's height to its shadow length (at the same time of day)
  • Example: A $6$ ft person casts a $4$ ft shadow, and a nearby tree casts a $30$ ft shadow. The height of the tree is $\frac{6}{4} \cdot 30 = 45$ ft
• Maps and scale drawings: The ratio of distances on a map or scale drawing is proportional to the ratio of actual distances
  • Example: On a map with a scale of $1$ inch to $10$ miles, a $2.5$-inch line represents an actual distance of $2.5 \cdot 10 = 25$ miles
• Photography: The ratio of the dimensions of an original image to its enlargement or reduction is the scale factor
• Architecture and design: Similar figures and proportions are used to create aesthetically pleasing and functional structures
  • Example: The Golden Ratio ($\approx 1.618:1$) is often used in art, architecture, and design for its perceived beauty and balance

Common Mistakes and How to Avoid Them

• Confusing ratios and fractions: Remember that ratios compare two quantities with the same units, while fractions represent a part of a whole
• Incorrectly setting up proportions: Make sure that corresponding terms are in the same position (numerator or denominator) in both ratios
• Forgetting to check for similarity conditions: Verify that the necessary congruence or proportionality conditions are met before applying similarity theorems
• Misapplying scale factors: Remember that lengths are multiplied by the scale factor, areas by the square of the scale factor, and volumes by the cube of the scale factor
• Neglecting to simplify ratios or proportions: Simplify ratios and proportions when possible to make calculations easier and avoid errors
• Not double-checking results: Always review your work to ensure that your answer makes sense in the context of the problem

Practice Problems and Solutions

1. If the ratio of the length to the width of a rectangle is $5:3$, and the perimeter is $64$ units, find the length and width.

   • Let the length be $5x$ and the width be $3x$. The perimeter is $2(5x + 3x) = 64$. Solve for $x$: $16x = 64$, so $x = 4$. The length is $5 \cdot 4 = 20$ units, and the width is $3 \cdot 4 = 12$ units.

2. In a right triangle, one leg is $3$ units longer than the other, and the hypotenuse is $15$ units. Find the lengths of the legs.

   • Let the shorter leg be $x$. The longer leg is $x + 3$, and the hypotenuse is $15$. Use the Pythagorean theorem: $x^2 + (x + 3)^2 = 15^2$. Expand and solve the quadratic equation: $x = 6$ or $x = -9$. Since lengths cannot be negative, the shorter leg is $6$ units, and the longer leg is $6 + 3 = 9$ units.

3. A $12$ ft ladder leans against a wall, reaching $10$ ft up the wall. How far is the base of the ladder from the wall?

   • This forms a right triangle with the wall as one leg, the ground as the other leg, and the ladder as the hypotenuse. Use the Pythagorean theorem: $10^2 + x^2 = 12^2$, where $x$ is the distance from the base of the ladder to the wall. Solve for $x$: $x = \sqrt{12^2 - 10^2} = \sqrt{44} \approx 6.63$ ft.

4. Triangle $ABC$ is similar to triangle $DEF$. If $AB = 6$, $BC = 8$, and $EF = 10$, find $DE$.

   • The scale factor is $\frac{EF}{BC} = \frac{10}{8} = \frac{5}{4}$. Since $AB$ corresponds to $DE$, $DE = \frac{5}{4} \cdot AB = \frac{5}{4} \cdot 6 = 7.5$.

5. In a triangle, the angle bisector of the largest angle divides the opposite side into segments of length $4$ and $9$. Find the lengths of the other two sides.

   • By the Angle Bisector Theorem, $\frac{4}{9} = \frac{x}{y}$, where $x$ and $y$ are the lengths of the other two sides. Let $y = 9k$. Then $x = 4k$, and the perimeter is $4k + 9k + 13 = 13(k + 1)$. The perimeter must be less than $13 \cdot 2 = 26$ since $13$ is the longest side. Choosing $k = 1$ gives side lengths of $4$, $9$, and $13$.