Question

Count the number of triangles and squares in the given figure.

 

• A. 21 triangles, 7 squares
• B. 18 triangles, 8 squares
• C. 20 triangles, 8 squares
• D. 22 triangles, 7 squares

Hint:

When counting geometric figures in puzzles, break down the figure into smaller sections and count individual shapes within those sections. Don't forget to include overlapping or combined shapes.

Answer 1

The Correct Option is A

Step 1: Count the triangles. - The figure consists of several small triangles within larger triangles. - There are 6 small triangles in the bottom layer, 3 triangles in the middle, and 2 triangles at the top of the pyramid. - Additionally, there are 10 triangles formed by combining different segments, including the larger triangles and overlapping areas. Thus, the total number of triangles is: \[ 6 + 3 + 2 + 10 = 21. \] Step 2: Count the squares. - The squares are present in the middle portion of the figure, formed by intersections of lines. - There are 7 squares in total. Thus, the total number of squares is: \[ \boxed{7}. \] Step 3: Final Answer. The figure contains 21 triangles and 7 squares. \[ \boxed{21\ \text{triangles},\ 7\ \text{squares}} \]

Answer 2

Recount the triangles by orientation instead of by layer, and check the total against each option.

1. Option (a) 21 triangles, 7 squares: Grouping the triangles as upward-pointing and downward-pointing pieces across the figure gives \(13\) upward-pointing and \(8\) downward-pointing triangles, totalling \(21\); the intersecting lines in the middle band form exactly \(7\) squares — matches.
2. Option (b) 18 triangles, 8 squares: under-counts the triangles formed by overlapping segments and over-counts the squares — rejected.
3. Option (c) 20 triangles, 8 squares: misses one composite triangle and double-counts a square — rejected.
4. Option (d) 22 triangles, 7 squares: counts one non-existent triangle from a false overlap — rejected.

Recounting by orientation confirms \(21\) triangles and \(7\) squares.

the correct answer is 21 triangles, 7 squares.