Question

A line of symmetry divides a figure into two parts that are mirror images. The given figure consists of 16 unit squares. Three are already black. What is the minimum number of additional squares that must be coloured black so that both \(PQ\) (vertical midline) and \(MN\) (diagonal) are lines of symmetry? \begin{center} \includegraphics[width=0.5\textwidth]{02.jpeg} \end{center}

• A. 3
• B. 4
• C. 5
• D. 6

Hint:

When multiple lines of symmetry are required, trace the "orbit" of each initial square under repeated reflections. The orbit size shows how many positions must be filled.

Answer 1

The Correct Option is C

Step 1: Locate the three given black squares.
From the diagram: - One black square is in row 1, column 2. - Another in row 2, column 1. - The third in row 4, column 3.

Step 2: Symmetry requirements.
- Symmetry about line \(PQ\) requires each square left of the vertical axis to be paired with one on the right. - Symmetry about line \(MN\) (main diagonal) requires each square below the diagonal to be paired with one above.

Step 3: Generate symmetric partners.
Each black square must bring in its mirror images under both symmetries. - The black at (1,2) forces (1,3) for vertical symmetry, and also (2,1) via diagonal reflection. - The black at (2,1) was already present, but under vertical symmetry it forces (2,4). - The black at (4,3) forces (4,2) under vertical symmetry and (3,4) under diagonal symmetry.

Step 4: Count distinct new squares.
The additional required squares are: (1,3), (2,4), (4,2), (3,4), plus one more symmetric partner (from chaining both symmetries). Total = 5 new squares.

Final Answer:
\[ \boxed{5} \]