Question

A line of symmetry is defined as a line that divides a figure into two parts in a way such that each part is a mirror image of the other part about that line. 
The given figure consists of 16 unit squares arranged as shown. In addition to the three black squares, what is the minimum number of squares that must be coloured black, such that both PQ and MN form lines of symmetry? (The figure is representative)

• A. 3
• B. 4
• C. 5
• D. 6

Hint:

- To satisfy symmetry, the arrangement of squares on one side of the line must be mirrored on the other side.
- For diagonal lines of symmetry, this typically means ensuring that elements along the line are symmetrically arranged.

Answer 1

The Correct Option is C

Step 1: Understand the Problem Setup
The figure consists of 16 unit squares arranged in a square shape. There are already 3 black squares. We are tasked with finding the minimum number of additional black squares to be coloured so that two specific lines (PQ and MN) act as lines of symmetry. A line of symmetry divides a shape into two identical mirror-image halves, meaning that every square on one side of the line must have a corresponding matching square on the other side.
Step 2: Identify the Lines of Symmetry
- The line \( PQ \) and the line \( MN \) are given as the symmetry lines.
- To achieve symmetry, the squares on one side of these lines must be mirrored on the other side.
Step 3: Examine the Figure's Symmetry Requirements
Looking at the arrangement of the squares, we note that symmetry is required both along the line \( PQ \) (diagonal from top-left to bottom-right) and along the line \( MN \) (another diagonal from bottom-left to top-right). For both of these lines to be symmetry axes, each black square on one side of these lines needs a corresponding black square on the other side.
- The black squares in the given figure are arranged in such a way that the initial three black squares already fulfill some symmetry requirements. However, there are still missing squares that need to be coloured to ensure that both \( PQ \) and \( MN \) are symmetry axes.
- By carefully observing the figure, we can determine that a minimum of 5 additional squares must be coloured to ensure this symmetry.
Step 4: Conclusion
Thus, the minimum number of squares that must be coloured black to satisfy the symmetry condition is 5.