Show two figures are similar by describing a sequence of rigid motions and a dilation

Learning goal: Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations and dilations, find the scale factor, and, given two similar figures with or without coordinates, describe a sequence that exhibits the similarity.

Dilation, stated with its conditions. A dilation has a center and a positive scale factor k. It sends each point along the ray from the center through the point to k times its original distance from the center. The center itself does not move. If k is greater than 1 the image is an enlargement; if k is between 0 and 1 it is a reduction; if k = 1 nothing moves. A dilation keeps every angle and multiplies every length by k, so it keeps the shape and changes the size. With k ≠ 1 it is not a rigid motion, because it changes lengths (with k = 1 it is the identity and changes nothing), but it shares with rigid motions the property of taking lines to lines and parallel lines to parallel lines. With center at the origin the rule is (x, y) → (kx, ky).

The definition of similar. Two figures are similar when one can be taken onto the other by a sequence of rotations, reflections, translations and dilations. Congruent figures are the special case in which the dilation has k = 1 or is not needed at all; every pair of congruent figures is also similar. Because each motion in the sequence keeps angles, and only the dilation changes lengths, and it changes all of them by the same factor, similar figures have equal corresponding angles and corresponding side lengths in one common ratio. That ratio is the scale factor from the first figure to the second. The factor in the reverse direction is its reciprocal: if the first-to-second factor is 1/2, the second-to-first factor is 2.

How to find the sequence. First match corresponding vertices, using angles (a right angle matches a right angle) and the relative lengths of sides (longest to longest). Second, compute the scale factor from one pair of corresponding sides and confirm that every other pair gives the same factor; if one pair gives a different number, the figures are not similar and no sequence exists. Third, dilate the first figure by that factor. Fourth, the dilated figure is now congruent to the second figure, so describe a sequence of rigid motions taking it onto the second, exactly as in the congruence lesson: check orientation, reflect if necessary, translate and rotate. Finally apply the whole sequence to every vertex and confirm that each lands on its match.

A worked model with coordinates. Triangle ABC has A(2, 2), B(6, 2), C(2, 5). Triangle DEF has D(−1, 1), E(−1, 3), F(−2.5, 1). Match the vertices. In ABC the right angle is at A, where the horizontal side AB meets the vertical side AC; AB = 4 and AC = 3, so the hypotenuse BC is 5, the diagonal of a 4-by-3 box. In DEF the right angle is at D: DE is vertical, from height 1 to height 3, length 2, and DF is horizontal, from x = −1 to x = −2.5, length 1.5. The hypotenuse EF is the diagonal of a 1.5-by-2 box, length 2.5. So D matches A, E matches B (the 2-unit side matches the 4-unit side) and F matches C.

Scale factor from ABC to DEF: DE/AB = 2/4 = 1/2, DF/AC = 1.5/3 = 1/2, EF/BC = 2.5/5 = 1/2. All three agree, so the figures can be similar with k = 1/2.

Step 1: dilate with center the origin and k = 1/2, rule (x, y) → (x/2, y/2). A(2, 2) → (1, 1), B(6, 2) → (3, 1), C(2, 5) → (1, 2.5). Call this triangle A₁B₁C₁. Its sides are 2, 1.5 and 2.5, the same as DEF, so it is congruent to DEF. Step 2: compare orientation. Walking A₁ → B₁ → C₁ goes right, then up-and-left: counterclockwise. Walking D → E → F goes up, then down-and-left: also counterclockwise. No reflection is needed. The horizontal side A₁B₁ must become the vertical side DE, so rotate 90° counterclockwise about the origin, rule (x, y) → (−y, x): A₁(1, 1) → (−1, 1) = D; B₁(3, 1) → (−1, 3) = E; C₁(1, 2.5) → (−2.5, 1) = F. Every vertex lands. The sequence "dilate from the origin by 1/2, then rotate 90° counterclockwise about the origin" exhibits the similarity.

Order. In this example the rotation could be done first: A(2, 2) → (−2, 2) → (−1, 1), the same D. That works because both the dilation and the rotation are centered at the origin. A translation does not commute with a dilation centered at the origin: the point (2, 0) dilated by 2 and then moved 1 right lands at (5, 0), but moved 1 right and then dilated by 2 lands at (6, 0). Describe the sequence in the order you used, and check with that order.

Without coordinates. A 5 cm by 8 cm photograph is printed at 15 cm by 24 cm and hung sideways. The factor is 15/5 = 3 and also 24/8 = 3, so the prints are similar. A sequence: dilate the small print by 3 from one of its corners, then rotate 90° about that corner so its sides run the same ways as the large print's, then translate it along the segment from that corner to the matching corner of the large print. Any measured pair of corresponding lengths must give the same 3. Two triangular tiles, one with sides 3, 4, 5 cm and the other 6, 8, 10 cm, are similar with factor 2; a triangle with sides 6, 8, 11 cm is not similar to either: against the 3, 4, 5 tile the ratios 6/3 and 8/4 are 2 but 11/5 is 2.2, and against the 6, 8, 10 tile the ratios 6/6 and 8/8 are 1 but 11/10 is 1.1.

Non-examples and the misconception. Rectangles of 3 by 5 and 6 by 9 have equal angles, all right angles, but 6/3 = 2 while 9/5 = 1.8, so they are not similar; no dilation multiplies one side by 2 and the other by 1.8. Students sometimes check only one pair of sides or only the angles. For rectangles the angles are always equal, so the angle check says nothing; for general figures a single side ratio can agree by coincidence. Every pair of corresponding sides must share the ratio, and every pair of corresponding angles must agree.

Worked enlargement with a rotation: Triangle with vertices (0, 0), (4, 0), (0, 2) and triangle with vertices (0, 0), (0, 6), (−3, 0). The legs are 4 and 2 in the first, 6 and 3 in the second; the factor is 6/4 = 3/2 and 3/2 = 3/2, consistent. Dilate from the origin by 3/2, rule (1.5x, 1.5y): (0, 0), (6, 0), (0, 3). Then rotate 90° counterclockwise about the origin, (−y, x): (6, 0) → (0, 6) and (0, 3) → (−3, 0). Both nonzero vertices land, and the origin stays; the sequence exhibits the similarity.

Worked non-example: A 3 by 5 rectangle and a 6 by 9 rectangle. Matching short side to short side, the ratio is 6/3 = 2; matching long to long, 9/5 = 1.8. The ratios differ, so no dilation followed by rigid motions can take one onto the other. Equal angles are not enough for similarity when the side ratios disagree.

Worked reduction with a translation: Triangle (2, 2), (6, 2), (6, 4) and triangle (5, −1), (7, −1), (7, 0). The legs are 4 and 2 in the first and 2 and 1 in the second; factor 1/2 both times. Dilate from the origin by 1/2: (1, 1), (3, 1), (3, 2). From (1, 1) to (5, −1) is 4 right and 2 down, so translate by (x + 4, y − 2): (1, 1) → (5, −1), (3, 1) → (7, −1), (3, 2) → (7, 0). All three land; sequence confirmed.

Worked, no coordinates: A 5 cm by 8 cm print and a 15 cm by 24 cm print hung sideways. Both length ratios are 3, so the prints are similar with factor 3. Dilate the small print by 3 from its bottom-left corner, rotate it 90° clockwise about that corner, so that its 24 cm side runs the same way as the large print's 24 cm side, then translate it along the segment from that corner to the matching corner of the large print (the corner where the large print's 15 cm and 24 cm sides meet in the same arrangement). A ruler check of any pair of corresponding edges must give the same factor 3.

Recording task: Draw triangle ABC with A(2, 2), B(6, 2), C(2, 5) and triangle DEF with D(−1, 1), E(−1, 3), F(−2.5, 1) on one grid. Record a table with columns: pre-image vertex, vertex after the dilation by 1/2 from the origin, vertex after the 90° counterclockwise rotation about the origin. Beneath it record the three side ratios DE/AB, DF/AC and EF/BC and show that each equals 1/2. Then trace A₁B₁C₁ and turn the tracing a quarter turn about the origin to confirm it lands on DEF.

Independent performance: Take triangle with vertices (1, 1), (5, 1), (1, 4). Make one enlargement by a dilation from the origin with k = 2 followed by a reflection across the x-axis, and one reduction by a dilation with k = 1/2 followed by a translation of your choice. Record both images as coordinate lists only, then, starting from the lists, describe a sequence that exhibits each similarity as though you did not know how the image was made. Success criteria: the scale factor is computed from all three side pairs and they agree; the dilation and each rigid motion are fully described with rules; the sequence is applied to every vertex and every vertex lands; the reverse scale factor is stated; and one sentence explains why a triangle with sides 4, 3, 6 could not be similar to the original.