NIMCET 2007 — Reasoning PYQ

1. Counting the Total Number of Squares

Let us break down the concentric structural design of the figure to count the squares first:

• Outer Structure (Large Square Grid):

  The main outer boundary forms a large square divided symmetrically by a horizontal and a vertical line through the center. This creates a standard $2\times 2$ grid:

  • Small individual $1\times 1$ squares = $4$

  • Main large $2\times 2$ boundary square = $1$

    $$\text{Outer Squares}=4+1=5$$

• Inner Structure (Small Diamond Square):

  Inside the main grid, there is an inner square tilted at an angle (appearing like a diamond grid) that connects the midpoints of the outer lines. This inner region is also divided symmetrically into four quarters:

  • Small individual $1\times 1$ inner squares = $4$

  • Main inner $2\times 2$ boundary square = $1$

    $$\text{Inner Squares}=4+1=5$$

• Total Squares:

  $$\text{Total Squares}=5+5=10\text{ squares}$$

2. Counting the Total Number of Triangles

To count the triangles efficiently, we use the standard trick where a square cross-cut by both diagonals forms exactly $8$ triangles (derived from $4\text{ small individual triangles}\times 2$).

• Triangles in the Inner Diamond Structure:

  The inner tilted square has both diagonal lines passing completely through its center. Applying the standard counting shortcut for a diagonal-crossed square:

  $$\text{Inner Triangles}=4\times 2=8$$

• Triangles in the Outer Square Structure:

  Now, look at the main outer square framework. It is also completely cut through by its two main diagonals, extending from corner to corner. Applying the same shortcut:

  $$\text{Outer Main Triangles}=4\times 2=8$$

• Triangles Formed by the Intersections (Middle Regions):

  Let's look at the intermediate spaces created between the outer square perimeter and the inner diamond boundary lines. Each of the $4$ main quadrants contains a group of smaller combined triangles:

  • In each corner/quadrant region, there are $2$ small triangles sharing an edge.

  • Since there are $4$ such regions around the symmetrical layout:

    $$\text{Intersection Triangles}=4\times 3=12$$

• Total Triangles:

  Summing up all the discovered components:

  $$\text{Total Triangles}=8+8=16\text{ (from major bases)}+12\text{ (from intermediate lines)}=28\text{ triangles}$$

Summary of Total Count

• Number of Triangles: $28$

• Number of Squares: $10$

This corresponds directly to the combination given in option (c).

Final Answer

The correct option is (c).