Understanding the Triangle Properties
To find the area of a right-angled triangle given the circumradius (R) and inradius (r), we can use the following relationships:
- The circumradius \( R \) of a right-angled triangle is given by the formula:
\[ R = \frac{c}{2} \]
where \( c \) is the length of the hypotenuse.
- The area \( A \) of the triangle can also be expressed in terms of the inradius \( r \):
\[ A = r \cdot s \]
where \( s \) is the semi-perimeter of the triangle.
Calculating Area with Given Values
1. Given Values:
- Circumradius \( R = 8.5 \)
- Inradius \( r = 3 \)
2. Finding the Hypotenuse:
- From the circumradius formula:
\[ c = 2R = 2 \times 8.5 = 17 \]
3. Using the Inradius to Find the Area:
- We know \( A = r \cdot s \).
- For a right-angled triangle, the semi-perimeter \( s \) can be expressed as:
\[ s = \frac{a + b + c}{2} \]
where \( a \) and \( b \) are the other two sides.
4. Area Formula:
- The area \( A \) can also be expressed as:
\[ A = \frac{1}{2} \times a \times b \]
5. Relating Area, Inradius, and Semi-perimeter:
- From the inradius relation, we can equate the two area formulas:
\[ A = r \cdot s \]
\[ A = \frac{1}{2} \times a \times b \]
6. Calculating the Area:
- The area can be calculated directly as:
\[ A = r \cdot s = 3 \cdot s \]
Using the properties of right triangles and the circumradius, you can derive the semi-perimeter and ultimately find:
\[ A = 3s \]
In conclusion, substituting the values will yield the area of the right-angled triangle.