Decide whether two figures are similar using similarity transformations, and explain why similar triangles have equal angles and proportional sides

Learning goal: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar, and explain, using similarity transformations, the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Similarity transformations. A similarity transformation is a sequence of rigid motions (translations, reflections, rotations) and dilations, in any order. Two figures are similar when some similarity transformation takes the first exactly onto the second. That is the definition, and every other fact about similar figures is a consequence of it. Congruent figures are a special case: a sequence of rigid motions with no dilation, or with a dilation of scale factor 1, is also a similarity transformation.

Each ingredient has known effects. Rigid motions preserve every length and every angle. A dilation with scale factor k leaves every angle unchanged, because each line goes to a parallel line, and multiplies every length by k. Put them together and a similarity transformation preserves every angle and multiplies every length by the product of the scale factors of the dilations used, which we call the scale factor of the similarity.

Deciding with coordinates: a worked model. Triangle ABC has A(0, 0), B(4, 0), C(0, 3). Triangle DEF has D(2, 1), E(2, 7), F(−2.5, 1). Are they similar? First, gather evidence that decides which vertices should correspond. In ABC the right angle is at A, where the horizontal leg AB = 4 meets the vertical leg AC = 3, and the hypotenuse BC = √(16 + 9) = 5. In DEF the right angle is at D, where the vertical segment DE (from height 1 to height 7, length 6) meets the horizontal segment DF (from x = 2 to x = −2.5, length 4.5), and EF = √(6² + 4.5²) = √(36 + 20.25) = √56.25 = 7.5. So D corresponds to A, E to B (the longer leg to the longer leg) and F to C. The three ratios DE/AB = 6/4, DF/AC = 4.5/3 and EF/BC = 7.5/5 are all 1.5, so a scale factor of 1.5 is the only candidate.

Now produce the transformation. Step 1, dilate from the origin by 1.5: A → (0, 0), B → (6, 0), C → (0, 4.5). The sides are now 6, 4.5 and 7.5, the same as DEF. Step 2, the leg that was horizontal must become vertical, so rotate 90° counterclockwise about the origin, rule (x, y) → (−y, x): (0, 0) → (0, 0), (6, 0) → (0, 6), (0, 4.5) → (−4.5, 0). Step 3, translate by (x + 2, y + 1): (0, 0) → (2, 1) = D, (0, 6) → (2, 7) = E, (−4.5, 0) → (−2.5, 1) = F. Every vertex lands, so the sequence dilate, rotate, translate is a similarity transformation taking ABC onto DEF, and the triangles are similar by definition. Writing the ratios was not the proof; the transformation was. The ratios told us what to try.

Deciding without coordinates. Two figures drawn on paper are similar exactly when one can be traced, enlarged or reduced from some center by a single factor, and then slid, turned or flipped onto the other. If a single pair of corresponding lengths gives a different ratio from another pair, no dilation can fix both, and if a single pair of corresponding angles differs, nothing in a similarity transformation can change an angle, so the figures are not similar. Rectangles 2 by 3 and 4 by 5 have all angles equal, but 4/2 = 2 while 5/3 ≈ 1.67, and the other pairing (2 with 5, 3 with 4) gives 2.5 and about 1.33, so no correspondence works and no similarity transformation exists. A square of side 2 and a rhombus of side 2 with a 60° angle have all sides in the ratio 1, but a 90° angle cannot be sent to a 60° angle, so they are not similar either. For figures with more than three sides, both checks are needed.

What similarity means for triangles. The standard asks for an explanation, in terms of transformations, of why "similar triangles" is the same thing as "all corresponding angles equal and all corresponding sides proportional". There are two directions.

First direction: if two triangles are similar, then the angles correspond equally and the sides proportionally. A similarity transformation takes the first triangle onto the second. The rigid motions in it change no angle and no length. The dilations change no angle and multiply every length by the same factor. So each angle of the second triangle equals the matching angle of the first, and each side of the second is k times the matching side of the first, which is what proportional means: the three ratios all equal k. In the model, the angles at A and D are both 90°, the angle at B is the angle whose tangent is 3/4, about 36.87°, and so is the angle at E, and the angles at C and F are both about 53.13°.

Second direction: if all corresponding angles are equal and all corresponding sides are in one ratio k, then the triangles are similar. Dilate the first triangle by k from any center. The dilated triangle has sides equal to the second triangle's sides, one pair at a time, because each side was multiplied by k. Two triangles with three pairs of equal sides are congruent, so a sequence of rigid motions takes the dilated triangle onto the second one. Dilation followed by those rigid motions is a similarity transformation, so the triangles are similar. The angle condition was not even needed in this direction for triangles, since three sides determine a triangle; it is needed for quadrilaterals and beyond, as the square and rhombus show.

The misconception. Students often decide similarity from a picture: "they look the same shape". Looking is where to start, not where to finish. Two triangles with angles 40°, 60°, 80° and 40°, 65°, 75° look alike but are not similar, because a rigid motion or dilation cannot turn 60° into 65°. The decision is made by checking corresponding parts or by exhibiting the transformation, and a written answer must include one of those.

Worked, similar by a reflection and dilation: Triangle PQR with P(0, 0), Q(6, 0), R(0, 2) and triangle STU with S(0, 0), T(−3, 0), U(0, 1). Legs 6 and 2 against 3 and 1: both ratios are 1/2. Dilate from the origin by 1/2: (0, 0), (3, 0), (0, 1). Reflect across the y-axis, (x, y) → (−x, y): (0, 0) → S, (3, 0) → (−3, 0) = T, (0, 1) → U. Every vertex lands; the triangles are similar with scale factor 1/2 from PQR to STU.

Worked, not similar: Rectangle with vertices (0, 0), (2, 0), (2, 3), (0, 3) and rectangle with vertices (0, 0), (4, 0), (4, 5), (0, 5). All eight angles are right angles. The short sides give 4/2 = 2 and the long sides give 5/3 ≈ 1.67. A dilation multiplies every length by one factor, and rigid motions change no length, so no similarity transformation produces both ratios. Not similar, despite equal angles.

Worked, sides proportional but not similar: A square with side 2 and a rhombus with side 2 whose angles are 60° and 120°. Every side ratio is 1. But a similarity transformation preserves angles, and a 90° angle cannot become 60°. Not similar. This is why, for figures other than triangles, both the angle condition and the side condition are checked.

Worked, explaining the meaning for triangles: Triangles JKL and MNO are similar, with a similarity transformation consisting of a dilation by 3 and a rotation. Then angle K = angle N, because neither a dilation nor a rotation changes an angle; and MN = 3 · JK, NO = 3 · KL, OM = 3 · LJ, because the rotation changes no length and the dilation multiplies each length by 3. So all corresponding angles are equal and all corresponding sides are in the ratio 3.

Recording task: Draw triangle ABC with A(0, 0), B(4, 0), C(0, 3) and triangle DEF with D(2, 1), E(2, 7), F(−2.5, 1) on one grid. Record a table with columns: vertex of ABC, after the dilation by 1.5 from the origin, after the 90° counterclockwise rotation about the origin, after the translation (x + 2, y + 1), and the matching vertex of DEF. Beneath it record the ratios DE/AB = 6/4, DF/AC = 4.5/3 and EF/BC = 7.5/5 as decimals, and the three pairs of corresponding angles (90° and 90°; about 36.9° and 36.9°; about 53.1° and 53.1°).

Independent performance: Take the triangle with vertices (1, 1), (7, 1), (1, 5). Make an image by a dilation from the origin with k = 1/2 followed by a reflection across the x-axis followed by a translation of your choice, and record only its three vertices. Then, starting from the two coordinate lists as though you did not know the history, decide whether the triangles are similar by exhibiting a similarity transformation and checking every vertex. Also write a paragraph, in terms of transformations, explaining why the two triangles must have three pairs of equal angles and three pairs of sides in one ratio, and give the ratio. Finally give one pair of quadrilaterals that are not similar although their angles all agree, with the two side ratios that disagree. Success criteria: the scale factor is computed from all three side pairs; each transformation is fully described with a rule; every vertex is checked; the explanation names what rigid motions preserve and what a dilation changes; and the non-example shows which condition failed.