The radius of a circle is 13 cm and a chord's distance from center is 12 cm, then the length of the chord is equal to :

Question

Accepted Answer

To determine the length of the chord in a circle with a given radius and distance from the center, we can use the Pythagorean theorem. Here's a detailed explanation supporting Option 3 as the correct answer:

1. **Understanding the Problem:**
   - We have a circle with a radius of 13 cm.
   - A chord is at a distance of 12 cm from the center of the circle.
   - We need to find the length of the chord.

2. **Applying the Pythagorean Theorem:**
   - Consider a right triangle formed by the radius, the distance from the center to the chord, and half of the chord.
   - The radius of the circle is the hypotenuse of this right triangle, which is 13 cm.
   - The distance from the center to the chord is one leg of the triangle, which is 12 cm.
   - Let the half-length of the chord be $ x $.

According to the Pythagorean theorem:
   $$
   x^2 + 12^2 = 13^2
   $$
   $$
   x^2 + 144 = 169
   $$
   $$
   x^2 = 169 - 144
   $$
   $$
   x^2 = 25
   $$
   $$
   x = 5
   $$

3. **Calculating the Full Length of the Chord:**
   - Since $ x $ is half of the chord, the full length of the chord is $ 2x $.
   - Therefore, the length of the chord is $ 2 \times 5 = 10 $ cm.

4. **Conclusion:**
   - The length of the chord is 10 cm, which corresponds to Option 3.

5. **Why Other Options are Incorrect:**
   - **Option 1 (15 cm):** This would imply a different set of values that do not satisfy the Pythagorean theorem with the given radius and distance.
   - **Option 2 (12 cm):** This is the distance from the center to the chord, not the length of the chord itself.
   - **Option 4 (20 cm):** This length is too long given the constraints of the radius and the distance from the center, and does not satisfy the Pythagorean theorem.

Thus, based on the calculations and logical reasoning, Option 3 (10 cm) is indeed the correct answer.