Transformations and Similarity - Grade 8

1. Similar and Congruent Figures

Congruent Figures:

Definition: Figures with the same size and shape

All corresponding sides are equal
All corresponding angles are equal
Symbol: \( \cong \)
Created by rigid transformations (translations, reflections, rotations)

Similar Figures:

Definition: Figures with the same shape but not necessarily the same size

Corresponding angles are equal
Corresponding sides are proportional (in the same ratio)
Symbol: \( \sim \)
Created by dilations or combinations of rigid transformations and dilations

Key Differences:

Property	Congruent	Similar
Size	Same size	May differ in size
Shape	Same shape	Same shape
Angles	Equal	Equal
Sides	Equal	Proportional

Important: All congruent figures are similar, but not all similar figures are congruent!

2. Dilations: Graph the Image & Find Coordinates

Definition: A dilation is a transformation that produces an image that is the same shape as the original, but is a different size.

Key Components:

Center of dilation: Fixed point from which all points are enlarged or reduced
Scale factor (k): Ratio of the new distance to the original distance
Enlargement: Scale factor \( k > 1 \)
Reduction: Scale factor \( 0 < k < 1 \)
No change: Scale factor \( k = 1 \)

Dilation Rule (Center at Origin):

\( (x, y) \rightarrow (kx, ky) \)

where \( k \) = scale factor

Properties of Dilations:

Angle measures are preserved (stay the same)
Orientation is preserved
Lines remain lines; parallel lines stay parallel
Side lengths are multiplied by the scale factor
NOT a rigid transformation (doesn't preserve distance)

Examples:

Example 1: Dilate point \( A(3, 4) \) by scale factor \( k = 2 \) with center at origin.

Rule: \( (x, y) \rightarrow (2x, 2y) \)

\( A(3, 4) \rightarrow A'(6, 8) \)

Example 2: Dilate point \( B(-6, 9) \) by scale factor \( k = \frac{1}{3} \) with center at origin.

Rule: \( (x, y) \rightarrow \left(\frac{1}{3}x, \frac{1}{3}y\right) \)

\( B(-6, 9) \rightarrow B'(-2, 3) \)

Example 3: Triangle with vertices \( C(2, 1) \), \( D(4, 3) \), \( E(1, 5) \) is dilated by \( k = 3 \). Find the new coordinates.

\( C'(6, 3) \), \( D'(12, 9) \), \( E'(3, 15) \)

3. Dilations: Find the Scale Factor

Formula to Find Scale Factor:

\( k = \frac{\text{Image length}}{\text{Pre-image length}} \)

Or using coordinates: \( k = \frac{x'}x = \frac{y'}{y} \)

Methods to Find Scale Factor:

Method 1: Using Side Lengths

Identify corresponding sides
Divide image side by pre-image side
Simplify the ratio

Method 2: Using Coordinates

Identify corresponding vertices
Divide image coordinate by pre-image coordinate
Check that all coordinates give same ratio

Examples:

Example 1: Point \( P(4, 6) \) is dilated to \( P'(12, 18) \). Find the scale factor.

\( k = \frac{x'}{x} = \frac{12}{4} = 3 \)

Check: \( k = \frac{y'}{y} = \frac{18}{6} = 3 \) ✓

Scale factor: \( k = 3 \)

Example 2: A triangle with side 15 cm is dilated to a similar triangle with corresponding side 5 cm. Find the scale factor.

\( k = \frac{5}{15} = \frac{1}{3} \)

Scale factor: \( k = \frac{1}{3} \) (reduction)

Example 3: Point \( Q(10, -8) \) dilates to \( Q'(5, -4) \). Find the scale factor.

\( k = \frac{5}{10} = \frac{1}{2} \) or \( k = \frac{-4}{-8} = \frac{1}{2} \)

Scale factor: \( k = 0.5 \) (reduction by half)

4. Identify Similar Triangles

Definition: Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.

Similarity Notation:

\( \triangle ABC \sim \triangle DEF \)

Read as: "Triangle ABC is similar to triangle DEF"

Properties of Similar Triangles:

Corresponding angles are congruent (equal)
Corresponding sides are proportional
Ratio of perimeters equals the scale factor
Ratio of areas equals the square of the scale factor

Proportionality of Sides:

If \( \triangle ABC \sim \triangle DEF \), then:

\( \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k \)

where \( k \) is the scale factor

Example:

\( \triangle PQR \sim \triangle STU \). If \( PQ = 6 \), \( QR = 8 \), \( PR = 10 \), and \( ST = 9 \), find \( TU \) and \( SU \).

Step 1: Find scale factor: \( k = \frac{ST}{PQ} = \frac{9}{6} = 1.5 \)

Step 2: \( TU = QR \times k = 8 \times 1.5 = 12 \)

Step 3: \( SU = PR \times k = 10 \times 1.5 = 15 \)

5. Angle-Angle (AA) Criterion for Similar Triangles

AA Similarity Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Why Two Angles Are Enough:

Since the sum of angles in a triangle is always 180°, if two angles are equal, the third angle must also be equal.

Other Similarity Criteria:

SSS (Side-Side-Side) Similarity:

If the three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar.

\( \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} \)

SAS (Side-Angle-Side) Similarity:

If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar.

\( \frac{AB}{DE} = \frac{BC}{EF} \) and \( \angle B \cong \angle E \)

Examples:

Example 1: In \( \triangle ABC \), \( \angle A = 50° \) and \( \angle B = 60° \). In \( \triangle DEF \), \( \angle D = 50° \) and \( \angle E = 60° \). Are the triangles similar?

Yes! Two angles are equal, so by AA criterion, \( \triangle ABC \sim \triangle DEF \)

Example 2: Triangle 1 has angles 45°, 55°, 80°. Triangle 2 has angles 45°, 55°, 80°. Are they similar?

Yes! All three angles are equal (AAA criterion, which is equivalent to AA)

6. Side Lengths and Angle Measures of Similar Figures

Key Principles:

Corresponding angles: Always equal in similar figures
Corresponding sides: Always proportional in similar figures

Steps to Find Missing Measures:

Write the similarity statement to identify corresponding parts
For angles: Corresponding angles are equal
For sides: Set up a proportion using corresponding sides
Solve the proportion for the unknown

Examples:

Example 1: \( \triangle ABC \sim \triangle XYZ \). If \( AB = 6 \), \( XY = 9 \), and \( BC = 8 \), find \( YZ \).

Set up proportion: \( \frac{AB}{XY} = \frac{BC}{YZ} \)

\( \frac{6}{9} = \frac{8}{YZ} \)

Cross-multiply: \( 6 \cdot YZ = 9 \cdot 8 \)

\( 6 \cdot YZ = 72 \) → \( YZ = 12 \)

Example 2: Two similar rectangles. First has dimensions 4 by 6. Second has length 10. Find the width of the second rectangle.

\( \frac{4}{w} = \frac{6}{10} \)

Cross-multiply: \( 6w = 40 \)

\( w = \frac{40}{6} = \frac{20}{3} = 6\frac{2}{3} \)

Example 3: \( \triangle MNO \sim \triangle PQR \). If \( \angle M = 75° \), find \( \angle P \).

Answer: \( \angle P = 75° \) (corresponding angles are equal)

7. Determine if Two Figures are Similar: Justify Your Answer

Methods to Prove Similarity:

Method 1: Check Angles and Sides

Verify all corresponding angles are equal
Verify all corresponding sides are proportional

Method 2: Use Similarity Theorems

AA: Two angles equal
SSS: All three sides proportional
SAS: Two sides proportional and included angle equal

Method 3: Transformation Approach

Show that one figure can be mapped to another using dilations and rigid transformations

Examples:

Example 1: Triangle 1 has sides 3, 4, 5. Triangle 2 has sides 6, 8, 10. Are they similar? Justify.

Check ratios: \( \frac{6}{3} = 2 \), \( \frac{8}{4} = 2 \), \( \frac{10}{5} = 2 \)

Yes, they are similar.

Justification: All corresponding sides are proportional with ratio 2:1 (SSS Similarity)

Example 2: Two triangles have angles 40°-60°-80° and 40°-60°-80°. Are they similar?

Yes, they are similar.

Justification: All corresponding angles are equal (AA Similarity)

Example 3: Triangle 1 has sides 2, 3, 4. Triangle 2 has sides 4, 6, 9. Are they similar?

Check ratios: \( \frac{4}{2} = 2 \), \( \frac{6}{3} = 2 \), \( \frac{9}{4} = 2.25 \)

No, they are NOT similar.

Justification: The ratios are not all equal, so sides are not proportional

8. Similar Triangles and Indirect Measurement

Concept: Using similar triangles to find measurements that are difficult to measure directly.

Common Applications:

Finding heights of tall objects (trees, buildings)
Finding distances across rivers or canyons
Shadow problems
Mirror/reflection problems

Steps for Indirect Measurement:

Identify the similar triangles in the problem
Label known and unknown measurements
Set up a proportion using corresponding sides
Solve for the unknown measurement

Examples:

Example 1 (Shadow Problem): A 6-foot person casts a 4-foot shadow. At the same time, a tree casts a 30-foot shadow. How tall is the tree?

The triangles formed are similar (same sun angle)

Set up proportion: \( \frac{\text{person height}}{\text{person shadow}} = \frac{\text{tree height}}{\text{tree shadow}} \)

\( \frac{6}{4} = \frac{h}{30} \)

\( 4h = 180 \) → \( h = 45 \) feet

Example 2 (Mirror Problem): A person 5 feet tall stands 8 feet from a mirror on the ground. The mirror is 12 feet from a flagpole. If the person can see the top of the flagpole in the mirror, how tall is the flagpole?

Similar triangles are formed (angle of incidence = angle of reflection)

\( \frac{5}{8} = \frac{h}{12} \)

\( 8h = 60 \) → \( h = 7.5 \) feet

9. Find Missing Side Lengths in Proportional Triangles

Setting Up Proportions:

General form: \( \frac{\text{side 1}}{\text{corresponding side 1}} = \frac{\text{side 2}}{\text{corresponding side 2}} \)

Steps to Solve:

Identify the similarity statement
Set up a proportion with the unknown
Cross-multiply
Solve for the unknown
Check your answer

Examples:

Example 1: \( \triangle ABC \sim \triangle DEF \). Given: \( AB = 5 \), \( BC = 7 \), \( DE = 15 \). Find \( EF \).

\( \frac{AB}{DE} = \frac{BC}{EF} \)

\( \frac{5}{15} = \frac{7}{EF} \)

\( 5 \cdot EF = 15 \cdot 7 \)

\( 5 \cdot EF = 105 \) → \( EF = 21 \)

Example 2: Two similar triangles. First triangle has sides 4, 6, 8. Second triangle has one side equal to 10 that corresponds to the side of length 4. Find the other two sides.

Scale factor: \( k = \frac{10}{4} = 2.5 \)

Second side: \( 6 \times 2.5 = 15 \)

Third side: \( 8 \times 2.5 = 20 \)

Example 3: If \( \frac{x}{9} = \frac{12}{27} \), find \( x \).

Cross-multiply: \( 27x = 9 \cdot 12 \)

\( 27x = 108 \) → \( x = 4 \)

10. Area and Perimeter of Similar Figures

Key Formulas:

Perimeter Ratio: \( \frac{P_1}{P_2} = k \)

The ratio of perimeters equals the scale factor

Area Ratio: \( \frac{A_1}{A_2} = k^2 \)

The ratio of areas equals the square of the scale factor

Examples:

Example 1: Two similar triangles have scale factor 3:1. If the smaller triangle has perimeter 12 cm, find the perimeter of the larger triangle.

\( \frac{P_{\text{large}}}{P_{\text{small}}} = 3 \)

\( P_{\text{large}} = 12 \times 3 = 36 \) cm

Example 2: Two similar rectangles have scale factor 2. If the smaller rectangle has area 20 cm², find the area of the larger rectangle.

\( \frac{A_{\text{large}}}{A_{\text{small}}} = 2^2 = 4 \)

\( A_{\text{large}} = 20 \times 4 = 80 \) cm²

Example 3: Two similar triangles have areas 25 cm² and 100 cm². What is the scale factor?

\( \frac{A_1}{A_2} = k^2 \)

\( \frac{25}{100} = k^2 \) → \( \frac{1}{4} = k^2 \) → \( k = \frac{1}{2} \)

Scale factor: 1:2 or 0.5

Quick Reference: Similarity Formulas

Concept	Formula
Dilation (center at origin)	\( (x, y) \rightarrow (kx, ky) \)
Scale Factor	\( k = \frac{\text{image}}{\text{pre-image}} \)
Proportional Sides	\( \frac{a}{d} = \frac{b}{e} = \frac{c}{f} = k \)
Perimeter Ratio	\( \frac{P_1}{P_2} = k \)
Area Ratio	\( \frac{A_1}{A_2} = k^2 \)