Triangle ABC is dilated by scale factor k = −2 from the origin. Which statement correctly describes the image?
AThe image is congruent to the original — a scale factor of 2 means equal-sized figures
BThe image is twice as large and appears rotated 180° around the origin
CThe image is half as large and reflected over the x-axis
DThe image is congruent and reflected, since the negative sign creates a reflection without size change
A negative scale factor combines scaling and reflection through the center. |k| = 2 doubles each point's distance from the origin, so the image is twice as large — similar to, not congruent with, the original. The negative sign places every image point on the opposite side of O from its preimage, producing a 180° rotation effect. Option 0 confuses magnitude with congruence. Option 2 has the size backwards (|−2| = 2, not 1/2). Option 3 incorrectly claims congruence.
Question 2 Multiple Choice
Point P is 5 units from the center of dilation. After a dilation with k = 3/4, how far is P' from the center, and what happened to the angle measure at P?
AP' is 15/4 units from center; the angle at P decreased proportionally
BP' is 15/4 units from center; the angle at P is unchanged
CP' is 20/3 units from center; the angle at P is unchanged
DP' is 20/3 units from center; the angle at P increased proportionally
Distance from the center scales by |k|: 5 × 3/4 = 15/4. But dilations preserve ALL angle measures — this is the defining property that makes the image similar to the preimage. The common misconception is that scaling changes angles proportionally; it does not. Angles are invariant under dilation precisely because both coordinate axes scale by the same factor k, keeping the ratio that determines any angle constant.
Question 3 True / False
A dilation generally changes the position of a figure in the plane.
TTrue
FFalse
Answer: False
When k = 1, every point P maps to P' such that OP' = 1 · OP — meaning P' = P for every point. The figure is unchanged in position and size. So a dilation with k = 1 is the identity transformation. A dilation also does not change the position of any point that lies at the center of dilation (it maps to itself for any k). 'Always changes position' is too strong a claim.
Question 4 True / False
After a dilation, the sides of the image figure are parallel to the corresponding sides of the preimage.
TTrue
FFalse
Answer: True
Dilations preserve angle measures. Since the angle each side makes with any reference direction is an angle measure, corresponding sides maintain the same orientation — they are parallel (or collinear if the center lies on the side). This is a consequence of the uniform scaling of both coordinate axes: the direction of each side (rise over run) is unchanged by k.
Question 5 Short Answer
Why does a dilation produce a similar figure rather than a congruent one, and under what condition would the image be congruent to the preimage?
Think about your answer, then reveal below.
Model answer: A dilation scales all distances from the center by |k|, changing the figure's size while preserving all angle measures. Equal angles and proportional sides is the definition of similarity, not congruence. The image is congruent only when |k| = 1 — either k = 1 (identity, no change) or k = −1 (a point reflection through the center that preserves distances while reflecting through the center).
The key distinction is that congruence requires preserving distances AND angles, while similarity only requires preserving angles (distances may scale). A dilation with |k| ≠ 1 specifically changes the scale, which is why it produces similarity. This is why dilations are the 'missing transformation' in the similarity story: rigid motions give congruence; rigid motions plus dilation give the full family of similarity transformations.