Click image to open full size
Scale Factor & Similar Figures
Scale Factor and Proportional Similarity
Related Tools
Related Labs
Related Worksheets
Related Cheat Sheets
Similar figures have the same shape but not always the same size. In geometry, scale factor tells how much a figure is enlarged or reduced. This idea is important for maps, blueprints, models, photos, and coordinate transformations. When students understand scale factor, they can compare lengths, predict missing sides, and describe dilations accurately.
A dilation changes a figure by multiplying distances from a fixed center point by the same scale factor. Corresponding angles stay congruent, while corresponding side lengths change in the same ratio. On a coordinate plane, a dilation centered at the origin with scale factor k sends each point (x, y) to (kx, ky). This makes similar figures a powerful way to connect geometry, algebra, and real-world measurement.
Key Facts
- Similar figures have congruent corresponding angles and proportional corresponding side lengths.
- Scale factor = image length ÷ original length.
- If k > 1, the dilation is an enlargement.
- If 0 < k < 1, the dilation is a reduction.
- Dilation centered at the origin: (x, y) becomes (kx, ky).
- Perimeter scale factor = k, but area scale factor = k^2.
Vocabulary
- Similar figures
- Figures that have the same shape because their corresponding angles are equal and their corresponding side lengths are proportional.
- Scale factor
- The number used to multiply each length of an original figure to get the matching length of its image.
- Dilation
- A transformation that enlarges or reduces a figure from a fixed center point using a scale factor.
- Center of dilation
- The fixed point from which all points of a figure move closer or farther during a dilation.
- Corresponding parts
- Matching sides, angles, or vertices in two figures that are in the same relative position.
Common Mistakes to Avoid
- Using original length ÷ image length for scale factor when the problem asks from original to image. The correct ratio is image length ÷ original length.
- Assuming similar figures must face the same direction. Figures can still be similar after rotations, reflections, or translations if angles match and side lengths are proportional.
- Multiplying angles by the scale factor. Dilation changes lengths but keeps all angle measures the same.
- Using k instead of k^2 for area changes. If side lengths are multiplied by 3, the area is multiplied by 9.
Practice Questions
- 1 Triangle ABC has side lengths 4 cm, 6 cm, and 8 cm. Triangle A'B'C' is a dilation of triangle ABC with scale factor 2.5. Find the three side lengths of A'B'C'.
- 2 A point P(3, -5) is dilated from the origin by scale factor 4. What are the coordinates of P'?
- 3 Two quadrilaterals have matching angles, but their side length ratios are 2:3, 4:6, 5:8, and 6:9. Are the quadrilaterals similar? Explain why or why not.