Scale Factor Anchor Chart
Geometry & Proportions Reference
Scale Factor Anchor Chart
Key Relationship
Original × Scale Factor = New
The Golden Formula
Scale factor (\(k\)) is the constant ratio that compares the size of the New (Image) figure to the Original (Pre-Image).
Scale Factor (\(k\)) =
\(k = \frac{\text{New Length}}{\text{Original Length}}\)
✓ Keep units consistent! Always: New ÷ Old
What Are Corresponding Sides?
Matching Positions
Sides that occupy the same relative position in two similar geometric figures. You must match corresponding sides before calculating!
Original (\(\triangle ABC\)) 4 cm 3 cm
× 2 Scale
New Image (\(\triangle DEF\)) 8 cm 6 cm
Bottom: 4 cm ↔ 8 cm
Height: 3 cm ↔ 6 cm
Ratio: \(\frac{8}{4} = \frac{6}{3} = \mathbf{2}\)
Behavior of Scale Factor (\(k\))
Shape stays identical; size transforms
Enlargement \(k > 1\)
The new image is larger than the original pre-image.
Values Look Like
2, 3.5, \(\frac{5}{2}\), 150%
Improper Fraction
\(\frac{\text{Numerator}}{\text{Denominator}} > 1\)
Example: \(k = 3 \implies\) Every side is 3 times longer.
Reduction \(0 < k < 1\)
The new image is smaller than the original pre-image.
Values Look Like
\(\frac{1}{2}\), \(\frac{3}{4}\), 0.25, 40%
Proper Fraction
\(\frac{\text{Top}}{\text{Bottom}} < 1\)
Example: \(k = \frac{1}{4} \implies\) Every side is cut to one-fourth.
How-To Examples: Master the 3 Calculations
Step-by-Step Problem Solving
1
Find Scale Factor
Original side is 12 cm; New side is 18 cm.
Step 1: Set ratio \(k = \frac{\text{New}}{\text{Original}}\)
Step 2: Substitute \(k = \frac{18}{12}\)
Step 3: Simplify \(k = \frac{3}{2} = 1.5\)
Enlargement because \(1.5 > 1\)
2
Find Scaled Side
Original side is 20 in; Scale factor \(k = \mathbf{\frac{1}{4}}\).
Step 1: Formula \(\text{New} = \text{Orig} \times k\)
Step 2: Multiply \(\text{New} = 20 \times \frac{1}{4}\)
Step 3: Solve \(\text{New} = \mathbf{5\text{ in}}\)
Reduction because \(\frac{1}{4} < 1\)
3
Find Original Side
Scaled side is 24 m; Scale factor \(k = \mathbf{3}\).
Step 1: Formula \(\text{Orig} = \frac{\text{New}}{k}\)
Step 2: Divide \(\text{Orig} = \frac{24}{3}\)
Step 3: Solve \(\text{Orig} = \mathbf{8\text{ m}}\)
Divide or multiply by reciprocal!
Scale Factor Golden Rules & Common Traps Quick Checklist
Order Matters!
Always put \(\frac{\text{New}}{\text{Original}}\). Inverting gives the reverse transformation.
Angles Don't Change!
Corresponding angles stay congruent (equal). Only side lengths change.
Scale Factor of 1
If \(k = 1\), the figures are congruent (identical size and identical shape).
Area Rule (Bonus)
Perimeter changes by \(k\), but Area changes by \(k^2\)!
