An equilateral triangle of sides 3 inch each is given. How many equilateral…

When two similar plane figures have side lengths in ratio k:1, their areas are in ratio k2:1. For an equilateral triangle cut into unit triangles of the same shape, this area-scaling relationship directly gives the number of unit triangles that tile it.

1. The given triangle has side 3 inch and the unit triangles cut from it have side 1 inch, so the linear scaling ratio is 3:1.

2. By Heron's formula, for the side-3 triangle: semi-perimeter s = (3+3+3)/2 = 4.5, so its area = √(s(s-3)(s-3)(s-3)) = √(4.5 × 1.5 × 1.5 × 1.5) = (9 × √(3))/4 sq. inch.

3. For a unit (side-1) equilateral triangle: semi-perimeter s = 1.5, so its area = √(1.5 × 0.5 × 0.5 × 0.5) = √(3)/4 sq. inch.

4. Since the interior of the larger triangle is exactly tiled by non-overlapping unit triangles, the count equals the total area divided by one unit triangle's area: (9√(3)/4) ÷ (√(3)/4) = 9.

Independently, since area scales with the square of the linear ratio, 32 = 9 - the same result, confirming the tiling count without needing the area computation at all.