Question:medium

In a shooting competition, all the shooters should hit the letter space in which letter 'A' is written as shown on the target board. The target board is a large equilateral triangle with each side 12 cm, and inside it the letter A is marked out as a smaller equilateral triangular space with each side 3 cm. What is the probability that the shooter will hit that space?

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Compare the letter-A triangle's side to the whole board's side; for similar triangles the area ratio is the square of the side ratio.
Updated On: Jul 21, 2026
  • 1/16
  • 1/12
  • 1/8
  • 1/4
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The Correct Option is A

Solution and Explanation

Instead of using the side-ratio shortcut, calculate both triangle areas with the standard equilateral-triangle area formula and divide.

  1. 1/16: matches the direct area calculation worked out below.
  2. 1/12: would need the inner triangle to be relatively larger than 3 cm on a 12 cm board, which does not match the figure.
  3. 1/8: also too large a share for a 3 cm triangle inside a 12 cm triangle.
  4. 1/4: would be the ratio if the inner side were 6 cm, not the 3 cm given here.

Area of the big triangle: $ \dfrac{\sqrt{3}}{4}(12)^2 = 36\sqrt{3} $ square cm. Area of the letter-A triangle: $ \dfrac{\sqrt{3}}{4}(3)^2 = \dfrac{9\sqrt{3}}{4} $ square cm. Probability $ = \dfrac{9\sqrt{3}/4}{36\sqrt{3}} = \dfrac{9}{144} = \dfrac{1}{16} $. So the correct choice is 1/16.

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