ΔEFG and ΔHIJ are similar. Also, ∠ E = ∠ H and ∠ F = ∠ I. If 3EF = HI and FG = 9 cm, then IJ is equal to ____.

Correct — A. The stated angle equalities fix the correspondence: ∠E = ∠H and ∠F = ∠I, so E↔H, F↔I and G↔J.

Corresponding sides are therefore EF ⁄ HI = FG ⁄ IJ
Given 3EF = HI, so EF ⁄ HI = 1 ⁄ 3
Then FG ⁄ IJ = 1 ⁄ 3 with FG = 9 cm
IJ = 3 × 9 = 27 cm → option (a)

Why the others are wrong

• (b)
  9 cm — 9 cm simply repeats FG. That is the congruent case, but 3EF = HI makes ΔHIJ three times the size of ΔEFG.
• (c)
  18 cm — 18 cm applies a scale factor of 2. The question fixes the factor at 3, not 2, so 18 cm belongs to a different triangle.
• (d)
  3 cm — 3 cm divides by 3 instead of multiplying, reading 3EF = HI backwards. HI is the larger side, so IJ must exceed FG, not shrink below it.

Similar triangles have equal corresponding angles and corresponding sides in one constant ratio, the scale factor.

The hard part is never the arithmetic; it is deciding which side pairs with which. Here the question tells you outright: ∠E = ∠H and ∠F = ∠I, so the side between E and F pairs with the side between H and I, and the remaining sides FG and IJ pair too.

The scale factor is hidden inside the equation 3EF = HI. Read it as HI is three times EF, so every side of ΔHIJ is three times its partner in ΔEFG.

Key facts

• Corresponding angles equal and corresponding sides in a fixed ratio is what similarity means.
• The angle equalities name the correspondence: EF pairs with HI, and FG pairs with IJ.
• 3EF = HI makes ΔHIJ the larger triangle, with scale factor 3.
• A side ratio of 3 makes the ratio of areas 3² = 9.

Common traps

• Reading 3EF = HI as EF = 3HI and shrinking the triangle instead of enlarging it.
• Pairing FG with HJ because the letters look adjacent, ignoring the stated angle equalities.
• Applying the factor 3 to the area as well, when areas scale by 9.

No directly related past PYQ was found.