In right angled `triangle ABC`, if AB=AC, then value of `|R/r|` (value [X] denote the greatest integer of x)

Answer

To solve the problem, we need to find the value of \(|R/r|\) for a right-angled triangle \(ABC\) where \(AB = AC\). Here’s a step-by-step solution:

### Step 1: Understand the Triangle
Given that triangle \(ABC\) is a right-angled triangle with \(AB = AC\), we can denote the lengths of the sides as follows:
- Let \(AB = AC = a\)
- Let \(BC = b\)

### Step 2: Use Pythagorean Theorem
Since \(A\) is the right angle, we can apply the Pythagorean theorem:
\[ b^2 = a^2 + a^2 = 2a^2 \implies b = a\sqrt{2} \]

### Step 3: Identify the Sides
Now we have the sides of the triangle:
- \(AB = AC = a\)
- \(BC = b = a\sqrt{2}\)

### Step 4: Calculate the Area (\(Δ\))
The area \(Δ\) of triangle \(ABC\) can be calculated using the formula for the area of a triangle:
\[ Δ = \frac{1}{2} \times base \times height = \frac{1}{2} \times a \times a = \frac{a^2}{2} \]

### Step 5: Calculate the Semi-perimeter (\(S\))
The semi-perimeter \(S\) is given by:
\[ S = \frac{AB + AC + BC}{2} = \frac{a + a + a\sqrt{2}}{2} = \frac{2a + a\sqrt{2}}{2} = a\left(1 + \frac{\sqrt{2}}{2}\right) \]

### Step 6: Calculate the Circumradius (\(R\))
The circumradius \(R\) of a triangle can be calculated using the formula:
\[ R = \frac{abc}{4Δ} \]
Here, \(a = a\), \(b = a\), and \(c = a\sqrt{2}\):
\[ R = \frac{a \cdot a \cdot a\sqrt{2}}{4 \cdot \frac{a^2}{2}} = \frac{a^3\sqrt{2}}{2a^2} = \frac{a\sqrt{2}}{2} \]

### Step 7: Calculate the Inradius (\(r\))
The inradius \(r\) is given by:
\[ r = \frac{Δ}{S} = \frac{\frac{a^2}{2}}{a\left(1 + \frac{\sqrt{2}}{2}\right)} = \frac{a}{2\left(1 + \frac{\sqrt{2}}{2}\right)} \]

### Step 8: Calculate \(|R/r|\)
Now we need to find \(|R/r|\):
\[ \frac{R}{r} = \frac{\frac{a\sqrt{2}}{2}}{\frac{a}{2\left(1 + \frac{\sqrt{2}}{2}\right)}} = \sqrt{2} \cdot \left(1 + \frac{\sqrt{2}}{2}\right) \]
Simplifying:
\[ \frac{R}{r} = \sqrt{2} + 1 \]

### Step 9: Calculate the Greatest Integer Function
Now, we need to find the greatest integer value of \(\sqrt{2} + 1\):
\[ \sqrt{2} \approx 1.414 \implies \sqrt{2} + 1 \approx 2.414 \]
Thus, the greatest integer less than or equal to \(2.414\) is \(2\).

### Final Answer
The value of \(|R/r|\) is:
\[ \boxed{2} \]