A ABC is inscribed in a circle with centre o and radius 5 cm and AC is a diameter of the circle. If AB = 8 cm, then BC = ......... cm.

Options

A. 10
B. 8
C. 6
D. 15

Answer

To solve the problem, we will use the properties of circles and the Pythagorean theorem.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- The radius of the circle is 5 cm.
- Therefore, the diameter AC = 2 × radius = 2 × 5 cm = 10 cm.
- The length of AB is given as 8 cm.

2. **Understand the Triangle:**
- Since AC is the diameter of the circle, triangle ABC is a right triangle (by the Thales' theorem).
- In triangle ABC, AC is the hypotenuse, and AB and BC are the other two sides.

3. **Apply the Pythagorean Theorem:**
- According to the Pythagorean theorem, for a right triangle:
\[ AC^2 = AB^2 + BC^2 \]
- Substituting the known values:
\[ 10^2 = 8^2 + BC^2 \]

4. **Calculate the Squares:**
- Calculate \(10^2\) and \(8^2\):
\[ 100 = 64 + BC^2 \]

5. **Rearranging the Equation:**
- Now, isolate \(BC^2\):
\[ BC^2 = 100 - 64 \]
\[ BC^2 = 36 \]

6. **Find the Value of BC:**
- Taking the square root of both sides:
\[ BC = \sqrt{36} = 6 \text{ cm} \]

### Final Answer:
BC = 6 cm.