A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius r
cm as shown in the figure. The side of the dodecagon is d cm. All the triangles
(numbered 1 to 12) in the figure are used to form squares of side r cm and each
numbered triangle is used only once to form a square.
The number of squares that can be formed and the number of triangles required to
form each square, respectively, are:
Note: The figure shown is representative.
Options:
(A) 3; 4
(B) 4; 3
(C) 3; 3
(D) 3; 2
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.8)**
**1. Intuitive Physical Setup & Given Parameters**:
- A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius $r$.
- It is partitioned from the center into 12 identical isosceles triangles (numbered 1 to 12).
- Each triangle has two sides equal to radius $r$ and apex angle at the center:
$\theta = \frac{360^\circ}{12} = 30^\circ$
- All 12 triangles are to be used (each exactly once) to form squares of side $r$.
- We need to find: (1) the number of squares formed, and (2) the number of triangles required per square.
**2. Step-by-Step Geometric Derivation**:
1. **Area of Each Triangle**:
The area of an isosceles triangle with two sides of length $r$ and included angle $30^\circ$ is:
$A_{\text{triangle}} = \frac{1}{2} r^2 \sin(30^\circ) = \frac{1}{2} r^2 \left(\frac{1}{2}\right) = \frac{1}{4} r^2$
2. **Area of a Square of Side $r$**:
$A_{\text{square}} = r^2$
3. **Triangles Required to Form One Square**:
$\text{Number of triangles per square} = \frac{A_{\text{square}}}{A_{\text{triangle}}} = \frac{r^2}{\frac{1}{4}r^2} = 4$
4. **Number of Squares Formed**:
With 12 total triangles available:
$\text{Number of squares} = \frac{12}{4} = 3\text{ squares}$
5. Result: **3 squares; 4 triangles per square** (Option A).
**3. Comprehensive Options & Distractor Analysis**:
- **(A) 3; 4**: ✅ **CORRECT**. Area of each triangle is $r^2/4$; thus 4 triangles form one square of area $r^2$, yielding $12/4 = 3$ squares.
- **(B) 4; 3**: ❌ **INCORRECT**. Reverses the two quantities.
- **(C) 3; 3**: ❌ **INCORRECT**. Total triangles would be $3 \times 3 = 9 \neq 12$.
- **(D) 3; 2**: ❌ **INCORRECT**. Total triangles would be $3 \times 2 = 6 \neq 12$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: Equating areas provides immediate answers: $A_{\triangle} = \frac{1}{2}r^2\sin 30^\circ = \frac{r^2}{4}$. Thus 4 triangles assemble into area $r^2$, creating 3 squares from 12 triangles.
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