A right triangle ABC, right angled at A, is circumscribing a circle. If AB = 8 cm and BC = 6 cm, find the radius of the circle.

Answer

To find the radius of the circle that circumscribes the right triangle ABC (right-angled at A), we can follow these steps:

### Step 1: Identify the sides of the triangle
Given:
- \( AB = 8 \, \text{cm} \)
- \( AC = 6 \, \text{cm} \)

### Step 2: Use the Pythagorean theorem to find the hypotenuse
In a right triangle, the Pythagorean theorem states:
\[ BC^2 = AB^2 + AC^2 \]
Substituting the values:
\[ BC^2 = 8^2 + 6^2 \]
\[ BC^2 = 64 + 36 \]
\[ BC^2 = 100 \]
Taking the square root:
\[ BC = \sqrt{100} = 10 \, \text{cm} \]

### Step 3: Use the formula for the radius of the circumcircle
For a right triangle, the radius \( R \) of the circumcircle can be calculated using the formula:
\[ R = \frac{c}{2} \]
where \( c \) is the length of the hypotenuse. Here, \( c = BC = 10 \, \text{cm} \).

Substituting the value:
\[ R = \frac{10}{2} = 5 \, \text{cm} \]

### Conclusion
The radius of the circle that circumscribes triangle ABC is \( 5 \, \text{cm} \).

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