The distance of the chord of length 16 cm from the centre of the circle of diameter 20 cm is :

Question

Accepted Answer

To determine the distance of the chord from the center of the circle, we can use the properties of a circle and the Pythagorean theorem. Let's break down the problem:

1. **Understanding the Circle:**
   - The diameter of the circle is given as 20 cm. Therefore, the radius (r) of the circle is half of the diameter, which is 10 cm.

2. **Understanding the Chord:**
   - The chord length is given as 16 cm. A chord is a straight line connecting two points on the circumference of the circle.

3. **Applying the Pythagorean Theorem:**
   - To find the distance from the center of the circle to the chord, we can draw a perpendicular from the center of the circle to the midpoint of the chord. This perpendicular bisects the chord into two equal segments of 8 cm each (since the chord is 16 cm long).

4. **Forming a Right Triangle:**
   - We now have a right triangle where:
     - One leg is half the chord length, which is 8 cm.
     - The hypotenuse is the radius of the circle, which is 10 cm.
     - The other leg is the distance from the center of the circle to the chord, which we need to find.

5. **Using the Pythagorean Theorem:**
   - According to the Pythagorean theorem: $ r^2 = d^2 + (c/2)^2 $
   - Here, $ r = 10 $ cm, $ c/2 = 8 $ cm, and $ d $ is the distance we need to find.
   - Plugging in the values: $ 10^2 = d^2 + 8^2 $
   - Simplifying: $ 100 = d^2 + 64 $
   - Solving for $ d^2 $: $ d^2 = 100 - 64 = 36 $
   - Therefore, $ d = \sqrt{36} = 6 $ cm.

Thus, the distance of the chord from the center of the circle is 6 cm, which corresponds to Option 3.

**Why Other Options are Incorrect:**

- **Option 1 (2 cm):** If the distance were 2 cm, the calculation using the Pythagorean theorem would not satisfy the equation $ 10^2 = 2^2 + 8^2 $, as this would result in $ 100 \neq 4 + 64 $.

- **Option 2 (4 cm):** Similarly, if the distance were 4 cm, the equation $ 10^2 = 4^2 + 8^2 $ would result in $ 100 \neq 16 + 64 $.

- **Option 4 (8 cm):** If the distance were 8 cm, the equation $ 10^2 = 8^2 + 8^2 $ would result in $ 100 \neq 64 + 64 $.

Therefore, the only correct option that satisfies the conditions of the problem is Option 3, which is 6 cm.