Question

Generating an Explanation of Finding Scale Factors
When you have two shapes to compare on a coordinate plane, you can determine the scale factor, knowing that the transformation was a dilation. Generate instructions you would give another student to determine the scale factor.

Answer 1

Step-by-step instructions to find the scale factor when one shape is a dilation of another on the coordinate plane:

1. Identify corresponding points.
- Match each vertex of the preimage to the corresponding vertex of the image (labeling helps: A → A', B → B', …).

2. Find (or confirm) the center of dilation.
- If the center is given, use it.
- If not given, draw (or compute) the lines through pairs of corresponding points (A to A', B to B', …). Those lines should all meet at the center of dilation. The intersection point is the center C.

3. Compute distances from the center to a corresponding pair.
- Pick any corresponding pair, e.g. P (preimage) and P' (image).
- Find the distance CP and CP' (use the distance formula) OR find the vector from C to P and C to P' (subtract coordinates).

4. Compute the scale factor k.
- k = (distance from center to image) / (distance from center to preimage) = CP' / CP.
- If you used vectors, k = (C→P') / (C→P) (componentwise if C is the origin), and the components should all be multiplied by the same k.

5. Check your result.
- Use another pair of corresponding points to verify the same k.
- If k is negative, the image is on the opposite side of the center from the preimage (a dilation with a sign flips direction; most problems expect k > 0).
- If CP = 0 (a preimage vertex is the center), you cannot use that vertex to compute k; use a different vertex.

Quick examples:
- If center is origin and A = (2,1), A' = (6,3): k = 6/2 = 3 (also 3/1 = 3).
- If center is C = (1,1), P = (2,2) and P' = (4,4): vectors C→P = (1,1), C→P' = (3,3) so k = 3/1 = 3.

Common pitfalls:
- Using non-corresponding points.
- Forgetting to use the center as the reference for distances.
- Trying to use a point that is the center (distance zero) to compute k.
- Not checking that all coordinate/component ratios are consistent.

That’s it: find the center, measure CP and CP', then k = CP'/CP (verify with a second point).