What is the ratio of inradius to the circumradius of a right angled triangle?

Options

A. A) 2:3
B. B) `1:sqrt2`
C. C) 5:6
D. D) can't be determined

Answer

To find the ratio of the inradius (r) to the circumradius (R) of a right-angled triangle, we can follow these steps:

### Step 1: Understand the properties of a right-angled triangle
In a right-angled triangle, the sides can be denoted as \( a \), \( b \), and \( c \), where \( c \) is the hypotenuse. The inradius \( r \) and circumradius \( R \) can be calculated using specific formulas.

### Step 2: Calculate the inradius (r)
The formula for the inradius \( r \) of a triangle is given by:
\[ r = \frac{A}{s} \]
where \( A \) is the area of the triangle and \( s \) is the semi-perimeter.

For a right-angled triangle:
- The area \( A \) can be calculated as:
\[ A = \frac{1}{2} \times a \times b \]
- The semi-perimeter \( s \) is:
\[ s = \frac{a + b + c}{2} \]

### Step 3: Calculate the circumradius (R)
The circumradius \( R \) of a right-angled triangle can be calculated using the formula:
\[ R = \frac{c}{2} \]
where \( c \) is the length of the hypotenuse.

### Step 4: Find the ratio \( \frac{r}{R} \)
Now that we have both \( r \) and \( R \), we can find the ratio:
\[ \frac{r}{R} = \frac{\frac{A}{s}}{R} = \frac{\frac{\frac{1}{2} \times a \times b}{\frac{a + b + c}{2}}}{\frac{c}{2}} \]

### Step 5: Simplify the expression
Substituting \( R \):
\[ \frac{r}{R} = \frac{\frac{1}{2} \times a \times b}{\frac{a + b + c}{2} \times \frac{c}{2}} = \frac{a \times b}{(a + b + c) \times c} \]

### Step 6: Use the Pythagorean theorem
Since \( c = \sqrt{a^2 + b^2} \), we can substitute this into our ratio:
\[ \frac{r}{R} = \frac{a \times b}{(a + b + \sqrt{a^2 + b^2}) \times \sqrt{a^2 + b^2}} \]

### Conclusion
The ratio of the inradius to the circumradius of a right-angled triangle simplifies to:
\[ \frac{r}{R} = \frac{a \times b}{(a + b + \sqrt{a^2 + b^2}) \times \sqrt{a^2 + b^2}} \]

### Final Answer
For a right-angled triangle, the ratio of the inradius to the circumradius is:
\[ \frac{r}{R} = \frac{1}{2} \]