One number scales every side
Equal corners and sides in the same ratio. That ratio is the scale factor.
Enlarging a photograph
A photograph is 35 mm wide. The photographer enlarges it to 45 mm wide.
Every line in the photograph becomes longer in the print.
How do the lines in the photograph grow in the print?
What this lesson covers
The idea
Two polygons with the same number of sides are similar if their corresponding angles are equal and their corresponding sides are in the same ratio; this common ratio is the scale factor (Representative Fraction).
Enlarging a photograph
A photograph is 35 mm wide. The photographer enlarges it to 45 mm wide.
Every line in the photograph becomes longer in the print.
How do the lines in the photograph grow in the print?
- Every line is multiplied by the same number.
- Every line gets the same extra length added.
- Short lines grow more than long lines.
Print the photograph
The blue house is the photograph. The orange house is the print, made with the scale factor k.
The scale factor
Two polygons with the same number of sides are similar if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio. This ratio is the scale factor (or Representative Fraction).
Maps and blueprints are drawn with a scale factor. In the house, every side of the print was 2 times its matching side and every corner was equal. A scale factor above 1 enlarges. A scale factor below 1, like 1/2, reduces.
Notes
Two polygons with the same number of sides are similar if their corresponding angles are equal and their corresponding sides are in the same ratio. That ratio is the scale factor.
Check yourself
A photograph 35 mm wide is enlarged to 45 mm wide. A line in the photograph is 14 mm long. How long is the matching line in the enlargement, in mm?
Answer: 18
The scale factor is 45/35 = 9/7. So the line is 14 × 9/7 = 18 mm.
A rectangle is 4 cm by 2 cm. A similar rectangle is made with scale factor 3. How long is its longer side, in cm?
Answer: 12
4 × 3 = 12 cm. The shorter side is 2 × 3 = 6 cm.
Pentagon P is similar to pentagon Q. A side of P is 5 cm and the matching side of Q is 15 cm. What is the scale factor from P to Q?
Polygon Q is a reduced copy of polygon P with scale factor 1/2. A side of P is 10 cm. How long is the matching side of Q, in cm?
Answer: 5
10 × 1/2 = 5 cm. A scale factor below 1 makes the copy smaller.
- 1/3. That is the scale factor the other way round, from Q to P.
- 3 — correct. Yes! 15 ÷ 5 = 3. Every side of Q is 3 times the matching side of P.
- 10. That is the difference 15 − 5. A scale factor is a ratio: 15 ÷ 5.