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Similarity Through
Transformations

Learn how rotations, reflections, translations, and dilations can map one figure onto another to prove they are similar.

8th Grade Math

What Does Similar Mean?

Two figures are similar if they have the same shape but not necessarily the same size. Every angle stays the same, and every side is multiplied by the same scale factor.

The Four Transformations

Translation
Slides every point the same distance and direction (no size or shape change).
Rotation
Turns the figure around a fixed point by a given angle.
Reflection
Flips the figure over a line, creating a mirror image.
Dilation
Enlarges or shrinks the figure from a center point by scale factor \( k \).

Rigid vs Non-Rigid Motions

Rigid motions (translations, rotations, reflections) preserve both shape and size: they produce congruent figures. A dilation is non-rigid: it changes size but keeps the shape, producing similar figures.

The Similarity Rule

A B C 5 5 6 dilation \( k = 2 \) A' B' C' 10 10 12 ~

A dilation with scale factor \( k = 2 \) maps \( \triangle ABC \) to \( \triangle A'B'C' \). Every side doubles, every angle stays the same: the triangles are similar.

Understanding Scale Factor \( k \)

1
\( k > 1 \): The image is larger than the original (enlargement).
2
\( 0 < k < 1 \): The image is smaller than the original (reduction).
3
\( k = 1 \): The figures are the same size: they are congruent, which is a special case of similar.

Congruent vs Similar

Congruent (\( \cong \)) Similar (\( \sim \))
Shape Same Same
Size Same Can differ
Angles All equal All equal
Sides All equal Proportional (\( k \))
Transformations Rigid only Rigid + dilation
Scale factor \( k = 1 \) Any \( k > 0 \)

How to Describe a Similarity Sequence

1
Check angles: Verify corresponding angles are equal. If not, the figures are not similar.
2
Find the scale factor: Divide a side of the image by the corresponding side of the original: \( k = \frac{\text{image}}{\text{original}} \).
3
Identify rigid motions first: Determine if you need a translation, rotation, or reflection to align orientation and position.
4
Apply the dilation: State the center of dilation and scale factor \( k \).
5
Write the sequence: List transformations in order, e.g., "Translate, then dilate by \( k = 2 \) centered at the origin."

Example 1: Simple Dilation

ABCD A'B'C'D' Center O 90 120 30 40

Example 2: Translate + Dilate

P Q R 3 3 4 translate P' Q' R' 6 6 8 dilate \( k = 2 \)

Example 3: Reflect + Dilate

line of reflection E F G 4 4 5 E' F' G' 6 6 7.5

Does Order Matter?

The final result of a similarity sequence can depend on the order you apply the transformations. However, for describing similarity, any valid sequence that maps one figure onto the other is acceptable.

Approach 1
Dilate first, then translate to align position.
Approach 2
Translate to align first, then dilate from the new position.

Identifying Corresponding Parts

When two figures are similar, each vertex, angle, and side in the original has a matching partner in the image. The order of letters tells you the correspondence.

A B C ~ D E F

When Figures Are Not Similar

If no combination of rigid motions and dilations can map one figure onto the other, the figures are not similar. This happens when:

1
Corresponding angles are not equal. Different shapes cannot be similar.
2
Side ratios are not consistent. If \( \frac{AB}{DE} \neq \frac{BC}{EF} \), the proportion breaks and \( k \) is not constant.

Common Mistakes

✕ Confusing Similar and Congruent
Saying figures are congruent when they only share shape, not size. Congruent requires \( k = 1 \).
✕ Forgetting to Check All Ratios
Checking only one pair of sides. You must verify that all corresponding side ratios equal the same \( k \).
✕ Omitting the Dilation
Describing only rigid motions (translate, rotate) without stating the dilation and its scale factor.

Summary of Transformations

Transformation Changes Size? Changes Shape? Type
Translation No No Rigid
Rotation No No Rigid
Reflection No No Rigid
Dilation Yes No Non-Rigid

Question 1 of 5

Which transformation is non-rigid and can change the size of a figure?

Question 2 of 5

\( \triangle ABC \) has sides 3, 4, 5. \( \triangle DEF \) has sides 9, 12, 15. What is the scale factor \( k \)?

Question 3 of 5

Figure A is mapped to Figure B by a reflection followed by a dilation with \( k = 0.5 \). Which statement is true?

Question 4 of 5

Rectangle JKLM has dimensions 2 by 6. Rectangle WXYZ has dimensions 3 by 12. Are they similar?

Question 5 of 5

\( \triangle PQR \sim \triangle STU \). If \( PQ = 8 \), \( ST = 12 \), and \( QR = 10 \), what is \( TU \)?