Triangles

In many triangle problems the key is understanding how a single geometric property scales. When two triangles are similar, every linear dimension—side lengths, medians, heights—changes by the same factor $k$. Because area depends on two dimensions, the area changes by the square of that factor, $k^2$. This simple relationship lets you move quickly between side ratios and area ratios, and it also helps when dealing with circles inscribed or circumscribed around triangles, because radii scale the same way as sides. Remember: *lengths* scale by $k$, *areas* scale by $k^2$. Use this rule to translate a given side ratio into an area ratio, or to find unknown lengths when you know how areas compare.