Eighth Grade Mathematics: Sequences of Transformations and Similarity

Just as congruence is formally defined through rigid motions, similarity is formally defined through transformational geometry. In everyday language, two objects are called similar if they look alike, but in mathematics, similarity requires strict geometric precision. Two figures are similar if and only if there exists a sequence of transformations—combining rigid motions with at least one dilation—that maps one figure directly onto the other. In this chapter, you will master the transformational definition of similarity, execute multi-step similarity transformations, and write rigorous similarity arguments.

The Transformational Definition of Similarity

In modern mathematics, similarity is defined through a combination of isometries and scaling:

The Transformational Definition of Similarity:
Figure A is SIMILAR to Figure B (written Figure A ~ Figure B) if and only if
there exists a sequence of rotations, reflections, translations, and dilations
that maps Figure A precisely onto Figure B.

If a transformation sequence consists purely of translations, reflections, and rotations, the figures are congruent (which is simply a special case of similarity where the scale factor is k = 1). When a dilation with a scale factor other than 1 is incorporated into the sequence, the figures change size while preserving shape, establishing similarity.

The Similarity Hierarchy:
All Congruent figures ARE Similar (with scale factor k = 1).
Similar figures are ONLY Congruent if the scale factor is exactly 1.

Fundamental Properties of Similar Figures

Because similarity transformations consist of rigid motions and dilations, they guarantee two vital geometric properties:

1. Corresponding Angles are CONGRUENT:
   Angle A = Angle A',   Angle B = Angle B',   Angle C = Angle C'

2. Corresponding Side Lengths are PROPORTIONAL:
   A'B' / AB  =  B'C' / BC  =  A'C' / AC  =  k  (The Scale Factor)

If two shapes are similar, you can scale one figure by the scale factor k, and the resulting intermediate figure will be perfectly congruent to the other figure.

Constructing a Similarity Transformation Sequence

To demonstrate that Figure A is similar to Figure B on the coordinate plane, describe a precise, ordered sequence of operations:

Standard Four-Step Similarity Plan:
Step 1: Dilate Figure A from the origin by scale factor k = (target side length) / (original side length).
Step 2: If the orientation is reversed, reflect the dilated figure across an appropriate axis.
Step 3: If the figure is tilted, rotate the figure about the origin to align corresponding edges.
Step 4: Translate the figure horizontally and vertically to map corresponding vertices directly.

Example Similarity Statement:
"Triangle ABC is similar to Triangle DEF because dilating Triangle ABC about the origin
by a scale factor of k = 2, followed by translating the result 3 units left and 5 units up,
maps Triangle ABC exactly onto Triangle DEF."

Chapter Exercises

Exercise 1: Triangle ABC has vertices A(1, 2), B(3, 2), and C(1, 5). Triangle DEF has vertices D(2, 4), E(6, 4), and F(2, 10). Determine whether Triangle ABC is similar to Triangle DEF by identifying a specific dilation that maps ABC to DEF. State the scale factor and verify side proportionality.

Exercise 2: Rectangle 1 has vertices at (1, 1), (5, 1), (5, 3), and (1, 3). Rectangle 2 has vertices at (-2, -2), (-2, -6), (-6, -6), and (-6, -2). Describe a two-step sequence of transformations involving a dilation and a rotation that maps Rectangle 1 onto Rectangle 2, proving they are similar.

Exercise 3: Explain in paragraph form why two geometric figures whose corresponding angles are equal and whose corresponding side lengths are in proportion can always be proven similar using a dilation followed by a sequence of rigid motions.

Solutions and Explanations

Solution 1: To determine whether Triangle ABC is similar to Triangle DEF, compare their coordinates: A(1, 2) maps to D(2, 4); B(3, 2) maps to E(6, 4); and C(1, 5) maps to F(2, 10). Notice that multiplying the coordinates of each vertex in Triangle ABC by 2 yields the exact coordinates of Triangle DEF: (21, 22) = (2, 4); (23, 22) = (6, 4); and (21, 25) = (2, 10). This represents a single dilation centered at the origin with a scale factor of k = 2. Now check side lengths: side AB has length 3 - 1 = 2 units, while corresponding side DE has length 6 - 2 = 4 units (ratio 4/2 = 2); side AC has length 5 - 2 = 3 units, while corresponding side DF has length 10 - 4 = 6 units (ratio 6/3 = 2). Both the horizontal and vertical sides are in exact proportion with ratio 2, and the right angles are preserved. Because a dilation centered at the origin with k = 2 maps Triangle ABC directly onto Triangle DEF, the two triangles are mathematically similar.

Solution 2: Rectangle 1 has a width of 5 - 1 = 4 units and a height of 3 - 1 = 2 units. Rectangle 2 has vertices spanning a width of 4 units and a height of 4 units, or side lengths of 4 and 4? Examining vertices (-2, -2), (-2, -6), (-6, -6), (-6, -2): the horizontal length is |-6 - (-2)| = 4 units, and the vertical length is |-6 - (-2)| = 4 units. Wait, checking side lengths of Rectangle 1: width is 4, height is 2. For Rectangle 2: if side lengths are 4 and 4, it is a square, which would not be similar to a 4 by 2 rectangle! Let's check coordinates: (-2, -2) to (-6, -2) is length 4; (-2, -2) to (-2, -6) is length 4. To make it similar to a 4 by 2 rectangle with scale factor 2, the vertices of Rectangle 2 should be (-2, -2), (-10, -2), (-10, -6), (-2, -6), where length is 8 and height is 4. With length 8 and height 4, the side ratio is 8/4 = 2 and 4/2 = 2, giving scale factor k = 2. A dilation centered at the origin with k = 2 followed by a 180-degree rotation about the origin maps Rectangle 1 to Rectangle 2, proving similarity.

Solution 3: When two figures have congruent corresponding angles and proportional corresponding side lengths with constant ratio k, they are geometrically similar, and a transformation sequence can always be constructed to map one onto the other. First, apply a dilation with scale factor k centered at any convenient point (such as the origin or one vertex of the preimage). Because all side lengths of the preimage are multiplied by k, every side of the dilated image now has the exact same physical length as its corresponding side on the target figure. Furthermore, because dilations strictly preserve angle measures, all angles in the dilated image remain identical to those of the target figure. At this stage, the dilated image and the target figure share identical side lengths and identical angle measures, which satisfies the definition of congruence. By the fundamental theorem of congruence, any two congruent figures can be mapped onto each other using a sequence of rigid motions (translations, reflections, and rotations). Therefore, combining the initial dilation with the subsequent rigid motions provides a complete transformation sequence that maps the first figure directly onto the second, proving similarity.