Proportions in triangles and scale factors
Parallel lines cut triangle sides in proportion, and scale factors affect areas and volumes in surprising ways. These ideas explain why giants could not exist.
The triangle proportionality theorem
If a line parallel to one side of a triangle meets the other two sides, it divides them proportionally. In triangle ABC with DE parallel to BC, D on AB, and E on AC, we get AD/DB = AE/EC. If AD = 4, DB = 6, and AE = 5, then EC = 7.5.
The midsegment theorem
A midsegment joins the midpoints of two sides of a triangle. It is parallel to the third side and half as long. If the third side is 18 cm, the midsegment is 9 cm. The three midsegments divide the triangle into four congruent triangles.
Angle bisectors and proportions
An angle bisector of a triangle divides the opposite side in the same ratio as the adjacent sides. If sides of 6 and 9 meet at the bisected angle, the opposite side is split in the ratio 6 : 9 = 2 : 3; for an opposite side of 10, the pieces are 4 and 6.
Areas of similar figures
If two figures are similar with scale factor k, their areas are in the ratio k squared. Doubling every length of a square multiplies its area by 4. A pizza 30 cm across has 2.25 times the area of one 20 cm across, since (30/20) squared = 2.25, which is why a large pizza is often better value.
Volumes of similar solids
For similar solids, volumes are in the ratio k cubed. A cube with edges twice as long holds 8 times as much. A model built at 1 : 10 scale has one hundredth of the surface area and one thousandth of the volume of the real object.
The square-cube law
Because volume, and so weight, grows with k cubed while strength, which depends on cross-sectional area, grows with k squared, simply scaling up an animal or structure makes it weaker relative to its weight. Galileo noted this in 1638: a giant ten times human height would weigh about 1,000 times more but have bones only about 100 times stronger, so its legs would break.
Working backwards from area
If two similar triangles have areas of 16 and 49 square units, the scale factor is the square root of 49/16, which is 7/4, so their perimeters are in the ratio 7 : 4. Take square roots of area ratios and cube roots of volume ratios to recover the length ratio.
Worked example
A photo 10 cm by 15 cm is enlarged by a scale factor of 3 to 30 cm by 45 cm. The perimeter triples, from 50 cm to 150 cm, but the area grows 9 times, from 150 to 1,350 square centimetres. Printing costs based on area rise much faster than the length increase suggests.
Key takeaways
A line parallel to one side of a triangle divides the other two sides proportionally; a midsegment is parallel to the third side and half its length. An angle bisector splits the opposite side in the ratio of the adjacent sides. For similar figures with scale factor k, lengths and perimeters scale by k, areas by k squared, and volumes by k cubed. The square-cube law, noted by Galileo in 1638, explains why scaled-up creatures would collapse. Recover length ratios with square roots of area ratios and cube roots of volume ratios.
Flashcard
Parallel line divides sides proportionally
Check your understanding
Two similar figures have a scale factor of 3. How do their areas compare?