Similar Triangles: Finding Side Length When Area Ratio is 1:16 and Larger Side is 42cm

Because area scales by the square of the linear scale factor! If sides are in ratio 1:4, then areas are in ratio $1^2:4^2 = 1:16$. So when given area ratio 1:16, the side ratio is $\sqrt{1}:\sqrt{16} = 1:4$.

Look at the area ratio! Since 16 > 1, the second triangle has the larger area. The side length of 42 cm belongs to this larger triangle, so we need to find the corresponding side in the smaller triangle.

Same process! Take square roots: $\sqrt{4}:\sqrt{9} = 2:3$. The side ratio would be 2:3, meaning corresponding sides are in the ratio 2:3.

Yes! Multiply your answer by the scale factor: $10.5 \times 4 = 42$ ✓. Also check that area scales correctly: if one triangle has 4 times the side length, it has $4^2 = 16$ times the area.

Similar triangles have the same shape but different sizes (like this problem). Congruent triangles have both same shape AND same size. In congruent triangles, corresponding sides are equal, not just proportional.