The ratio of areas of similar figures

When a triangle is enlarged by a factor k, its base becomes k times longer and its height becomes k times longer. The area is ½ × base × height, so it becomes ½ × (k × base) × (k × height) = k² × (the old area). Check with numbers: a triangle with base 6 and height 4 has area 12. Enlarge it with k = 5/2: base 15, height 10, area 75, and 12 × (5/2)² = 12 × 25/4 = 75 ✓. So for similar triangles, area ratio = k², where k is the ratio of any two corresponding sides. The altitudes to corresponding sides are also in the ratio k, because the small right triangles they form are similar. The same holds for corresponding medians and angle bisectors. Warning: the ratio of the areas is k², but the ratio of the perimeters is k. To go from the ratio of areas back to k, take the square root: area ratio 4 : 9 means k = 2 : 3.