Revise: Areas of Similar Triangles
How the area of similar triangles scales with the square of the side ratio, and how that contrasts with the (linear) scaling of perimeter.
Area₁/Area₂ = (side ratio)², not the side ratio itself.
k=3 → area ratio = 9.
Given an area ratio, take the square root to find the side ratio.
Areas 50, 98 → side ratio √(25/49)=5/7.
Perimeter scales linearly with the sides; only area scales squared.
Side ratio 2:5 → perimeter ratio 2:5 (not 4:25).
Proof idea: base and height both scale by k, so area (½×base×height) scales by k².
Sides 5,15 (k=3), small area 20 → large area 20×9=180.
Squaring goes side→area; square-rooting goes area→side.
Areas 36, 121 → sides in ratio 6:11.
Perimeter is a sum of sides (×k); area is a product of two lengths (×k²).
Side ratio 3:7, small perimeter 24 → large perimeter 56.