If the ratio of the areas of two similar triangles is 25 : 144, the ratio of their corresponding sides is:

Correct — B. Rule: in similar triangles the areas are in the ratio of the squares of corresponding sides.

So, writing the sides as s₁ and s₂:
(s₁ ⁄ s₂)² = 25 ⁄ 144

Take the square root of both sides:
s₁ ⁄ s₂ = √25 ⁄ √144 = 5 ⁄ 12

The corresponding sides are in 5 : 12 → option (b).

Why the others are wrong

• (a)
  12 : 5 — 12 : 5 is the right pair of numbers in the wrong order. The areas were given as 25 : 144, so the triangle of area 25 is named first, and its sides must be the shorter ones.
• (c)
  25 : 12 — 25 : 12 square-roots only the second term. Leaving 25 untouched compares an area against a length, which is not a ratio of anything. Both terms must be rooted.
• (d)
  5 : 144 — 5 : 144 square-roots only the first term. The missing step is √144 = 12; rooting one side of a ratio and not the other changes its value.

Similarity scales everything by the same factor k, but not to the same power.

Lengths — sides, perimeters, altitudes, medians, angle bisectors, radii of the inscribed and circumscribed circles — all scale by k.

Areas scale by k², because area is a length times a length. For similar solids, volumes scale by k³.

So an area ratio is always a square, and going from areas back to sides means taking a square root — of both terms.

Key facts

• Ratio of areas of similar triangles = (ratio of corresponding sides)².
• The ratio of perimeters of similar triangles equals the ratio of their sides.
• √25 = 5 and √144 = 12, so 25 : 144 is exactly the square of 5 : 12.
• For similar solids the ratio of volumes is the cube of the ratio of edges.

Common traps

• Reversing the order of the ratio so the larger triangle comes first
• Square-rooting one term of the ratio and not the other
• Using the area ratio unchanged as the ratio of perimeters

No directly related past PYQ was found.