Area ratio for similar rectangles – UGC NET Question

For similar rectangles, all linear dimensions scale in the same ratio. If lengths are in the ratio 5:8 and the widths also scale by the same factor, then the areas scale as the square of the linear ratio. Thus, A₁:A₂ = 5²:8² = 25:64. This uses the general principle that area is a two-dimensional measure and scales with the square of the similarity ratio.

Option A:
Option A, 5:8, would be correct for comparing lengths but not areas. It ignores that area involves two dimensions, and so it underestimates the extent to which the larger rectangle’s area exceeds the smaller’s.

Option B:
Option B, 25:64, is correct because it properly squares both terms in the length ratio. This preserves the similarity relationship while accurately reflecting the two-dimensional nature of area. No other option matches the squared ratio implied by geometric similarity.

Option C:
Option C, 10:16, simplifies back to 5:8 and therefore still represents only the linear ratio, not the area ratio. Area must scale as the square of the linear factor, so 10:16 is not correct for areas.

Option D:
Option D, 125:512, is the cube of 5:8 and would be appropriate for comparing volumes of similar three-dimensional solids, not areas of rectangles. Applying a cubic scaling factor to a two-dimensional quantity is conceptually wrong in this context.