Calculator
White themex,y (and optional label). We will compute P′ for each row using O(h,k) and k.
| # | Label | P(x,y) | P′(x′,y′) |
|---|
Export and Log
Computation Log
| # | Timestamp | Mode | Inputs | Result |
|---|
Example Data
| Case | Inputs | Action |
|---|---|---|
| Lengths | L = 7.5, L′ = 12 | |
| Coordinates | O(0,0), P(2,3), P′(3,4.5) | |
| Areas | A = 9, A′ = 36 | |
| Forward | O(1,−2), P(4,5), k = 1.2 | |
| Inverse | O(1,−2), P′(7.6,7.6), k = 1.2 | |
| Batch sample | Three points near O(0,0) |
Formula used
A dilation with center O(h,k) and scale factor k sends a point
P(x,y) to P′(x′,y′) given by:
x′ = h + k·(x − h)
y′ = k + k·(y − k)
Distances and perimeters scale by k. Areas scale by k².
- From lengths:
k = L′ / L - From coordinates:
k = d(O,P′) / d(O,P) - From areas:
k = √(A′ / A) - From perimeters:
k = P′ / P
How to use this calculator
- Choose a tab: find
k, mapP → P′, invert, batch, or graph. - Enter known values or upload data, then compute.
- Click Compute to see the result and steps.
- Use Add to Log to capture runs.
- Export your log as CSV for spreadsheets or PDF for sharing.
- Use the graph to visually confirm directions from the center.
- Batch mode accelerates classroom datasets and grading reviews.
- Raise precision for exact ratios; lower it for quick checks.
How to calculate the scale factor of a dilation
The scale factor compares “image size” to “original size.” Positive values preserve direction from the center; negative values flip direction (reflection through the center) while scaling by |k|.
- From two lengths: measure any corresponding segment before (
L) and after (L′). Computek = L′/L. Ifk > 1the figure enlarges; if0 < k < 1it shrinks. - From areas of similar figures: use
k = √(A′/A). Remember that areas scale withk². - From perimeters: use
k = P′/P. Perimeters scale linearly withk. - From coordinates with center
O(h,k):- Compute distances to the center:
k = d(O,P′)/d(O,P). - Sign rule: if
\overrightarrow{OP}and\overrightarrow{OP′}point the same way,kis positive; if opposite,kis negative.
- Compute distances to the center:
- Quick origin case: if the center is
(0,0)andP′ = (x′,y′) = k(x,y), then you may take any nonzero component:k = x′/x = y′/y(they must agree if points are correctly corresponding).
O(1,-2), P(4,5), and image P′(7.6,7.6). Then
d(O,P) = √((4−1)² + (5+2)²) = √(3²+7²) = √58 and
d(O,P′) = √((7.6−1)² + (7.6+2)²) = √(6.6²+9.6²) = √137.52,
so k = d(O,P′)/d(O,P) ≈ √137.52 / √58 ≈ 1.2. The rays OP and OP′ point the same way, so k is positive.
Note on symbols: to avoid clashing with the center’s y‑coordinate k in O(h,k), some texts use s or r for the scale factor. In this tool we label it k.
What does a negative scale factor mean?
A negative value reflects the image through the center and scales by |k|.
The ray OP′ points opposite OP.
- If
k = -1, every point maps to its opposite across the center. - Distances from the center are multiplied by
|k|. - Perimeters scale by
|k|; areas byk^2.
Scale factor from similarity and ratios
Similar figures share equal angle measures and proportional side lengths. Any corresponding side ratio equals the scale factor.
Choose the cleanest pair: k = L′/L = a′/a = b′/b. Verify with a second pair to validate correspondences.
Common mistakes and edge cases
- Zero distance: if
Pcoincides withO,kfrom distances is undefined. - Wrong center: using the origin when the center is not
(0,0)gives incorrectP′. - Mixing units: lengths must share units before forming ratios.
- Rounding drift: keep adequate precision or use exact fractions for proofs.
- Mismatched pairs: verify which sides/points actually correspond in the two figures.