ABC is triangle right angled at B. If AB = 6 cm and BC= 8 cm, then what is the length of the circum- radius of the `DeltaABC` ?

Options

A. 10 cm
B. 7 cm
C. 6 cm
D. 5 cm

Answer

To find the circumradius of triangle ABC, which is right-angled at B, we can follow these steps:

### Step 1: Identify the sides of the triangle
Given:
- AB = 6 cm (one leg)
- BC = 8 cm (the other leg)

### Step 2: Use the Pythagorean theorem to find the hypotenuse (AC)
The Pythagorean theorem states that in a right triangle:
\[ AC^2 = AB^2 + BC^2 \]

Substituting the values:
\[ AC^2 = 6^2 + 8^2 \]
\[ AC^2 = 36 + 64 \]
\[ AC^2 = 100 \]

Taking the square root to find AC:
\[ AC = \sqrt{100} = 10 \, \text{cm} \]

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

### Step 4: Calculate the circumradius
Substituting the value of the hypotenuse:
\[ R = \frac{10}{2} = 5 \, \text{cm} \]

### Final Answer
The length of the circumradius of triangle ABC is **5 cm**.

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