Q16
Find the number of triangles in the following figure :
Answer: D
Set up the figure with coordinates. The outer square has corners (0,0), (6,0), (6,6) and (0,6). The inner square occupies the bottom-right part, with corners (2,0), (6,0), (6,4) and (2,4).
Lines in the figure:
- The two diagonals of the outer square. The diagonal from (0,6) to (6,0) also passes through (2,4), so it serves as one diagonal of the inner square too.
- The inner square's other diagonal, from (2,0) to (6,4).
- The inner square's left side (x = 2) and top side (y = 4).
Count by region:
- Made from the big square and its diagonals (corners plus centre (3,3)): 8. These are the 4 quarters and the 4 half-squares.
- Inside the inner square, using its diagonals and centre (4,2): 8.
- Between the two squares: 5. These are (2,4)-(4,4)-(2,2), (2,4)-(4,4)-(3,3), (2,4)-(2,2)-(3,3), (4,4)-(6,4)-(6,6) and (0,0)-(2,0)-(2,2).
Total = 8 + 8 + 5 = 21. A full check of every set of three connected points gives 21 as well.
Answer: 21