If each side of an equilateral triangle is 37√3 cm, then its altitude (in cm) is equal to

Correct — D. The question, printed as an image, gives an equilateral triangle of side 37√3 cm and asks for the altitude.

The altitude of an equilateral triangle of side a is (√3⁄2)a — the height splits it into two 30-60-90 triangles.

With a = 37√3:
h = (√3⁄2) × 37√3
= 37 × (√3 × √3) ⁄ 2
= 37 × 3 ⁄ 2 = 111⁄2

h = 55.5 cm → option (d)

Why the others are wrong

• (a)
  37.5 — 37.5 keeps the side's coefficient 37 and attaches a half. No route through h = (√3⁄2)a reaches it: the √3 × √3 = 3 has to appear, and it makes the numerator 111.
• (b)
  18.5 — 18.5 is exactly 37 ⁄ 2 — the value you get by cancelling the two √3 factors to 1 instead of multiplying them to 3. Restore that 3 and 18.5 becomes 55.5.
• (c)
  60.5 — 60.5 is 121 ⁄ 2, so it needs a numerator of 11² from somewhere. The numerator here is 37 × 3 = 111, which gives 55.5.

An equilateral triangle has only one free number, its side a, and every other measurement follows from it.

Altitude is (√3⁄2)a and area is (√3⁄4)a². Both carry a √3, and here the side is itself a multiple of √3, so the surd cancels and the answer lands as a clean decimal.

A side of the form k√3 gives an altitude of 3k⁄2, so 37√3 gives 111⁄2 = 55.5 with no irrational left over.

Key facts

• The altitude of an equilateral triangle of side a is (√3⁄2)a.
• The area of an equilateral triangle of side a is (√3⁄4)a².
• A side written as k√3 gives an altitude of 3k⁄2, because √3 × √3 = 3.

Common traps

• Reaching for the area formula (√3⁄4)a² when the question asks for the altitude.
• Cancelling √3 × √3 to 1 rather than 3, which drops the answer to 18.5.

No directly related past PYQ was found.