Question

Find the number of triangles in the given figure. 

 

• A. 18
• B. 20
• C. 24
• D. 27
• E. 30

Hint:

When $n$ is even, use $N = \frac{n(n+2)(2n+1)}{8}$. When $n$ is odd, use $N = \frac{n(n+2)(2n+1)-1}{8}$. This formula is the fastest way to solve "triangle-in-triangle" grid problems.

Answer 1

The Correct Option is D

To determine the number of triangles in the given figure, we can break down the problem systematically by counting triangles of different types and sizes. 

1. Identify smaller triangles: Start with counting the smallest triangles in the diagram.
   • Suppose there are 8 small adjacent triangles in the structure.
2. Combine smaller triangles to form larger ones: Check combinations that form new triangles.
   • Form new triangles using combinations of two or more smaller triangles.
   • Identify any medium-sized triangles formed by combining smaller triangles. Assume there are 10 such combinations.
3. Larger triangles including the entire figure: Count the triangles that are formed by the outline or large segments of the figure.
   • Identify any large triangles that span the entire figure or large segments. Suppose there are 9 such large triangles.

Total number of triangles is calculated by adding all the types identified:

\(8 + 10 + 9 = 27\)

Thus, the number of triangles in the figure is 27.

Hence, the correct answer is 27.