Out of the two concentric circle the radius of the outer circle is `5` `cm` and the chord `AC` of the length `8` `cm` is a tangent to the inner circle find the radius of the inner circle.

Options

A. 2 cm
B. 3 cm
C. 4 cm
D. 5 cm

Explanation

To find the radius of the inner circle, we can follow these steps:

### Step 1: Understand the Problem
We have two concentric circles (one inside the other). The radius of the outer circle (OA) is given as 5 cm. A chord AC of the outer circle has a length of 8 cm and is tangent to the inner circle.

### Step 2: Identify Key Points
Let:
- O = Center of both circles
- A and C = Points on the outer circle such that AC is the chord
- D = Midpoint of chord AC
- T = Point where the tangent touches the inner circle

Since AC is a chord of the outer circle and is tangent to the inner circle, the line segment OD (from the center O to the midpoint D of the chord AC) is perpendicular to AC.

### Step 3: Calculate AD
Since D is the midpoint of AC, we can find the length of AD:
- Length of AC = 8 cm
- Therefore, AD = AC / 2 = 8 cm / 2 = 4 cm.

### Step 4: Apply the Pythagorean Theorem
In triangle OAD, we can apply the Pythagorean theorem:
- OA² = OD² + AD²
- We know OA = 5 cm and AD = 4 cm.

Substituting the known values:
\[ 5^2 = OD^2 + 4^2 \]
\[ 25 = OD^2 + 16 \]

### Step 5: Solve for OD
Now, we can solve for OD:
\[ OD^2 = 25 - 16 \]
\[ OD^2 = 9 \]
\[ OD = \sqrt{9} \]
\[ OD = 3 \text{ cm} \]

### Conclusion
The radius of the inner circle (OD) is 3 cm.

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