Scale Factor Anchor Chart
Geometry & Proportions Reference
Scale Factor Anchor Chart
Golden Relationship
Original × \(k\) = New
The Scale Factor Formula (\(k\))
Ratio: New ÷ Old
\(k = \frac{\text{New Length (Image)}}{\text{Original Length (Pre-Image)}}\)
Memory Trick: Think “N over O” (New / Old) ✓ Keep units identical
What is Scale Factor?
The constant multiplier used to enlarge or reduce a geometric shape proportionally.
• Multiplicative: Always multiply or divide—never add or subtract.
• Same Shape: Angles stay identical; sides stay proportional.
Corresponding Sides in Similar Figures
Sides must occupy the exact same relative position
Original (\(\triangle ABC\))
4 cm 3 cm
× 2 Scale Factor \(k = 2\)
New Image (\(\triangle DEF\))
8 cm 6 cm
Matching Bases:
\(k = \frac{8\text{ cm}}{4\text{ cm}} = \mathbf{2}\)
Matching Heights:
\(k = \frac{6\text{ cm}}{3\text{ cm}} = \mathbf{2}\)
Both ratios must be equal!
Enlargement \(k > 1\)
The new image is larger than the original pre-image.
Examples of \(k\) \(2,\ 3.5,\ \frac{5}{2},\ 150\%\)
Fraction Test \(\text{Numerator} > \text{Denominator}\)
Quick Check: Multiplying by a number > 1 increases size.
Reduction \(0 < k < 1\)
The new image is smaller than the original pre-image.
Examples of \(k\) \(\frac{1}{2},\ \frac{3}{4},\ 0.25,\ 40\%\)
Fraction Test \(\text{Numerator} < \text{Denominator}\)
Quick Check: Multiplying by a proper fraction decreases size.
How-To: Master the 3 Problem Types
Step-by-step solutions
1
Find Scale Factor
Original = 12 cm, New = 18 cm
1. Formula: \(k = \frac{\text{New}}{\text{Old}}\)
2. Substitute: \(k = \frac{18}{12}\)
3. Simplify: \(k = \frac{3}{2} = 1.5\)
Enlargement (\(1.5 > 1\))
2
Find Scaled (New)
Original = 20 in, Scale \(k = \mathbf{\frac{1}{4}}\)
1. Formula: \(\text{New} = \text{Old} \times k\)
2. Multiply: \(\text{New} = 20 \times \frac{1}{4}\)
3. Solve: \(\text{New} = \mathbf{5\text{ in}}\)
Reduction (\(\frac{1}{4} < 1\))
3
Find Original (Old)
New Image = 24 m, Scale \(k = \mathbf{3}\)
1. Formula: \(\text{Old} = \frac{\text{New}}{k}\)
2. Divide: \(\text{Old} = \frac{24}{3}\)
3. Solve: \(\text{Old} = \mathbf{8\text{ m}}\)
Reverse: Divide by \(k\)
Scale Factor Golden Rules & Common Pitfalls Anchor Reference
1. Order Matters
Always put \(\frac{\text{New}}{\text{Old}}\). Inverting gives the reciprocal scale factor.
2. Angles Stay Same
Corresponding angles are equal (congruent). Only edge lengths change!
3. If \(k = 1\)
The figures are congruent (same exact size and shape).
4. Area Rule (\(k^2\))
Perimeter scales by \(k\), but Area scales by \(k^2\)!
