The radius of a circle with centere P is 25 cm . The length of a chord of the same circle is 48 cm . Find the distance of the chord from the centre P of the circle.

Answer

To find the distance of the chord from the center \( P \) of the circle, we can follow these steps:

### Step 1: Understand the Geometry
We have a circle with center \( P \) and radius \( r = 25 \) cm. A chord \( AB \) of the circle has a length of \( 48 \) cm. We need to find the perpendicular distance from the center \( P \) to the chord \( AB \).

### Step 2: Bisect the Chord
Since the perpendicular from the center of the circle to the chord bisects the chord, we can find the length of \( AO \) (where \( O \) is the midpoint of the chord \( AB \)):
\[ AO = \frac{AB}{2} = \frac{48}{2} = 24 \text{ cm} \]

### Step 3: Set Up the Right Triangle
Now, we can visualize a right triangle \( PAO \) where:
- \( PA \) is the radius of the circle, which is \( 25 \) cm.
- \( AO \) is half the length of the chord, which is \( 24 \) cm.
- \( PO \) is the distance from the center \( P \) to the chord \( AB \), which we need to find.

### Step 4: Apply the Pythagorean Theorem
According to the Pythagorean theorem:
\[ PA^2 = PO^2 + AO^2 \]
Substituting the known values:
\[ 25^2 = PO^2 + 24^2 \]
Calculating the squares:
\[ 625 = PO^2 + 576 \]

### Step 5: Solve for \( PO^2 \)
Rearranging the equation to solve for \( PO^2 \):
\[ PO^2 = 625 - 576 \]
\[ PO^2 = 49 \]

### Step 6: Find \( PO \)
Taking the square root of both sides gives us:
\[ PO = \sqrt{49} = 7 \text{ cm} \]

### Conclusion
The distance of the chord from the center \( P \) of the circle is \( 7 \) cm.

---