In a circle O is the center and the length of diameter and one chord is 25 cm and 21 cm respectively. Then find the perpendicular distance from center to the chord.

Options

A. (A). `√46 cm`
B. (B). `√23 cm`
C. (C). `√21 cm`
D. (D). `√34 cm`

Answer

To find the perpendicular distance from the center of the circle to the chord, we can follow these steps:

### Step 1: Identify the given values
- The diameter of the circle is 25 cm.
- The length of the chord is 21 cm.

### Step 2: Calculate the radius of the circle
The radius (r) is half of the diameter. Therefore:
\[ r = \frac{\text{Diameter}}{2} = \frac{25 \text{ cm}}{2} = 12.5 \text{ cm} \]

### Step 3: Divide the chord into two equal parts
Since the perpendicular from the center of the circle to the chord bisects the chord, we can find the length of half the chord (BD):
\[ BD = \frac{\text{Length of chord}}{2} = \frac{21 \text{ cm}}{2} = 10.5 \text{ cm} \]

### Step 4: Apply the Pythagorean theorem
In the right triangle ODB, where:
- \( OB \) is the hypotenuse (the radius of the circle),
- \( BD \) is one leg (half the chord),
- \( OD \) is the other leg (the perpendicular distance we want to find).

According to the Pythagorean theorem:
\[ OB^2 = OD^2 + BD^2 \]

### Step 5: Substitute the known values into the equation
Substituting the values we have:
\[ (12.5)^2 = OD^2 + (10.5)^2 \]
Calculating the squares:
\[ 156.25 = OD^2 + 110.25 \]

### Step 6: Solve for \( OD^2 \)
Rearranging the equation to solve for \( OD^2 \):
\[ OD^2 = 156.25 - 110.25 = 46 \]

### Step 7: Calculate \( OD \)
Taking the square root to find \( OD \):
\[ OD = \sqrt{46} \text{ cm} \]

### Final Answer
The perpendicular distance from the center of the circle to the chord is:
\[ OD \approx 6.78 \text{ cm} \]

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