There is a circle of radius 5 cm. There are two chords AB and AC OF equal length = 6 cm . Find the length of chord BC .

Answer

To find the length of chord BC given the circle with radius 5 cm and two equal chords AB and AC of length 6 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Draw the Circle and Chords**:
- Draw a circle with center O and radius 5 cm.
- Draw two equal chords AB and AC, both measuring 6 cm.

2. **Identify the Midpoint**:
- Let P be the midpoint of chord BC. Since AB and AC are equal, the line segment AP bisects angle A, making BP equal to PC.

3. **Use Right Triangle Properties**:
- Drop a perpendicular from O to chord AB at point P. Since P is the midpoint, AP = 3 cm (half of 6 cm).
- In triangle OAP, we can apply the Pythagorean theorem:
\[ OA^2 = OP^2 + AP^2 \]
- Here, OA (radius) = 5 cm and AP = 3 cm. Thus:
\[ 5^2 = OP^2 + 3^2 \]
\[ 25 = OP^2 + 9 \]
\[ OP^2 = 25 - 9 = 16 \]
\[ OP = 4 \text{ cm} \]

4. **Calculate BP**:
- Now, we can find BP using the Pythagorean theorem in triangle OBP:
\[ OB^2 = OP^2 + BP^2 \]
- Here, OB (radius) = 5 cm and OP = 4 cm:
\[ 5^2 = 4^2 + BP^2 \]
\[ 25 = 16 + BP^2 \]
\[ BP^2 = 25 - 16 = 9 \]
\[ BP = 3 \text{ cm} \]

5. **Find Length of BC**:
- Since BP = PC, we have:
\[ BC = BP + PC = 3 + 3 = 6 \text{ cm} \]

### Final Answer:
The length of chord BC is **6 cm**.