The diameter and a chord of a circle have a common end point. If the lengt of the diameter is 20 cm and the length of the chord is 12 cm, how far is the chrod from the centre of the circle?

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Problem
We have a circle with a diameter and a chord that share a common endpoint. The length of the diameter is 20 cm, and the length of the chord is 12 cm. We need to find the distance from the center of the circle to the chord.

### Step 2: Draw the Diagram
1. Draw a circle.
2. Mark the center of the circle as point O.
3. Draw the diameter AB such that its length is 20 cm. Thus, OA = OB = 10 cm (since the radius is half of the diameter).
4. Mark point C on the circumference of the circle such that AC is the chord with a length of 12 cm.

### Step 3: Identify Key Points
- Let O be the center of the circle.
- A and B are the endpoints of the diameter.
- C is the endpoint of the chord.
- Let L be the foot of the perpendicular from O to the chord AC.

### Step 4: Calculate the Lengths
- The radius OA = 10 cm.
- The length of the chord AC = 12 cm.
- Since L is the midpoint of AC, we can find AL:
\[ AL = \frac{AC}{2} = \frac{12}{2} = 6 \text{ cm} \]

### Step 5: Apply the Pythagorean Theorem
In triangle OAL:
- OA is the hypotenuse (10 cm).
- AL is one leg (6 cm).
- OL is the other leg (the distance we need to find).

According to the Pythagorean theorem:
\[ OA^2 = OL^2 + AL^2 \]
Substituting the known values:
\[ 10^2 = OL^2 + 6^2 \]
\[ 100 = OL^2 + 36 \]
\[ OL^2 = 100 - 36 \]
\[ OL^2 = 64 \]
\[ OL = \sqrt{64} = 8 \text{ cm} \]

### Step 6: Conclusion
The distance from the center of the circle to the chord is 8 cm.

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