In the figure, given alongside, CD is a diameter which meets the chord AB at E, such that AE = BE = 4 cm. If CE is 3 cm, find the radius of the circle.

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step-by-Step Solution:

1. **Identify Given Information:**
- AE = BE = 4 cm (since E is the midpoint of AB)
- CE = 3 cm
- CD is the diameter of the circle.

2. **Define Variables:**
- Let the radius of the circle be \( r \).
- Since O is the center of the circle, we have \( OA = r \) and \( OC = r \).

3. **Use the Property of Chords:**
- Since OE is perpendicular to chord AB at point E (because OE bisects AB), we can use the Pythagorean theorem in triangle OAE.

4. **Find OE:**
- We know that \( OE = OC - CE \).
- Therefore, \( OE = r - 3 \) (since CE = 3 cm).

5. **Apply the Pythagorean Theorem:**
- In triangle OAE, we have:
\[ OA^2 = OE^2 + AE^2 \]
- Substituting the known values:
\[ r^2 = (r - 3)^2 + 4^2 \]

6. **Expand and Simplify:**
- Expanding \( (r - 3)^2 \):
\[ r^2 = (r^2 - 6r + 9) + 16 \]
- Simplifying the equation:
\[ r^2 = r^2 - 6r + 25 \]
- Canceling \( r^2 \) from both sides:
\[ 0 = -6r + 25 \]
- Rearranging gives:
\[ 6r = 25 \]
- Therefore:
\[ r = \frac{25}{6} \text{ cm} \]

7. **Convert to Mixed Fraction or Decimal:**
- Converting \( \frac{25}{6} \) to a mixed fraction:
\[ 25 \div 6 = 4 \text{ remainder } 1 \Rightarrow 4 \frac{1}{6} \text{ cm} \]
- As a decimal:
\[ r \approx 4.16 \text{ cm} \]

### Final Answer:
The radius of the circle is \( \frac{25}{6} \) cm or approximately \( 4.16 \) cm.

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