If `d_1,d_2,d_3` be the three distinct diagonals of a regular octagon, such that `d_1 lt d_2 lt d_3`, find the ratio of the inradius to the circumradius of the octagon.

Options

A. `d_1/d_3`
B. `sqrtd_2/d_3`
C. `d_2/d_3`
D. `d_1d_3/(d_2)^2`

Answer

To find the ratio of the inradius to the circumradius of a regular octagon, we can follow these steps:

### Step 1: Understand the Definitions
- **Inradius (r)**: The radius of the largest circle that can fit inside the octagon (the incircle).
- **Circumradius (R)**: The radius of the smallest circle that can encompass the octagon (the circumcircle).

### Step 2: Diagonals of the Octagon
In the context of the problem, we have three distinct diagonals of the octagon denoted as \(d_1\), \(d_2\), and \(d_3\) such that \(d_1 < d_2 < d_3\).

### Step 3: Relate Diagonals to Radii
From the problem statement, we can infer that:
- The inradius \(r\) corresponds to the smallest diagonal \(d_1\).
- The circumradius \(R\) corresponds to the largest diagonal \(d_3\).

### Step 4: Set Up the Ratio
The ratio of the inradius to the circumradius can thus be expressed as:
\[ \frac{r}{R} = \frac{d_1}{d_3} \]

### Step 5: Conclusion
Therefore, the ratio of the inradius to the circumradius of the octagon is:
\[ \frac{r}{R} = \frac{d_1}{d_3} \]

### Final Answer
The correct answer is option A: \( \frac{d_1}{d_3} \).

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