Chords AB and CD of a circle interest at E and are perpendicular to each other. Segments AE, EB and ED are of lengths 2 cm, 6 cm and 3 cm respectively. Then the length of the diameter the circle in cm is :

Options

A. `sqrt(65)`
B. `(1)/(2) sqrt(65)`
C. 65
D. `(65)/(2)`

Answer

To solve the problem step by step, we will use the information given about the chords AB and CD that intersect at point E and are perpendicular to each other.

### Step 1: Identify the lengths of segments
We have the following lengths:
- AE = 2 cm
- EB = 6 cm
- ED = 3 cm

### Step 2: Calculate the length of CE
Since the chords are perpendicular to each other, we can use the property of intersecting chords. The product of the segments of one chord is equal to the product of the segments of the other chord.

Using the formula:
\[ AE \times EB = CE \times ED \]

Substituting the known values:
\[ 2 \times 6 = CE \times 3 \]
\[ 12 = CE \times 3 \]

Now, solve for CE:
\[ CE = \frac{12}{3} = 4 \text{ cm} \]

### Step 3: Calculate the total length of chord CD
Since CE = 4 cm and ED = 3 cm, we can find the total length of chord CD:
\[ CD = CE + ED = 4 + 3 = 7 \text{ cm} \]

### Step 4: Find the midpoint of chord CD
Since the chords are perpendicular and intersect at E, the point E divides the chord CD into two equal parts. Therefore, each half of the chord CD is:
\[ \frac{CD}{2} = \frac{7}{2} = 3.5 \text{ cm} \]

### Step 5: Calculate the radius of the circle
Now, we can use the right triangle formed by the radius (OC), half of the chord (3.5 cm), and the distance from the center of the circle to the chord (OE). We can find OE using the Pythagorean theorem.

Let OC be the radius (R), then:
\[ OC^2 = OE^2 + CE^2 \]

We know:
- CE = 4 cm
- OE = 3.5 cm (half of CD)

Using the Pythagorean theorem:
\[ R^2 = OE^2 + CE^2 \]
\[ R^2 = \left(\frac{7}{2}\right)^2 + 4^2 \]
\[ R^2 = \left(\frac{49}{4}\right) + 16 \]
\[ R^2 = \frac{49}{4} + \frac{64}{4} \]
\[ R^2 = \frac{113}{4} \]

### Step 6: Calculate the radius
Taking the square root:
\[ R = \sqrt{\frac{113}{4}} = \frac{\sqrt{113}}{2} \]

### Step 7: Calculate the diameter of the circle
The diameter (D) is twice the radius:
\[ D = 2R = 2 \times \frac{\sqrt{113}}{2} = \sqrt{113} \]

### Final Answer
The length of the diameter of the circle is \( \sqrt{113} \) cm.

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