NIMCET 2007 — Reasoning PYQ

1. Understanding the Objective

A pentagon is any closed geometric shape bounded by exactly $5$ straight line segments (sides). In non-verbal reasoning problems like this one, pentagons are typically formed by combining smaller internal triangles and quadrilaterals inside the main rectangular grid.

2. Systematic Counting Method

Let us break down the symmetrical geometry of the figure provided in image_bdd7a9.png to identify the hidden pentagons:

• Outer Boundary Regions (Corner Pentagons):

  Look at each of the $4$ outer corners of the large main rectangle (Top-Left, Top-Right, Bottom-Left, Bottom-Right).

  By tracking the lines running from a corner to the inner diamond junctions, you will find a distinct $5\text{-sided}$ polygon at each corner.

  $$\text{Corner Pentagons}=4$$

• Side-Facing Central Pentagons (Edges):

  Now consider the perimeter edges of the main rectangle:

  • Along the top edge, there are two symmetric pentagons meeting at the top-center vertex.

  • Along the bottom edge, there are likewise two symmetric pentagons meeting at the bottom-center vertex.

  • Along the left edge, there is a large pentagon pointing inwards.

  • Along the right edge, there is another large pentagon pointing inwards.

  $$\text{Top Edge}=2$$

  $$\text{Bottom Edge}=2$$

  $$\text{Left Edge}=1$$

  $$\text{Right Edge}=1$$

  $$\text{Total Side-Facing Pentagons}=2+2+1+1=6$$

• Inner Cross / Large Combined Pentagons:

  By grouping the center diamond segments with the side diagonals, we can find larger overlapping pentagons. There are exactly $2$ pointing upwards/downwards and $4$ distributed symmetrically around the central intersection axis.

  $$\text{Inner/Large Pentagons}=6$$

3. Calculating Total Number of Pentagons

Summing up all the distinct groups of pentagons found within the figure:

$$\text{Total Pentagons}=4+6+6=16$$

Final Answer

There are exactly $16$ pentagons in the figure.

Therefore, the correct option is (c).