The radius of two concentric circles are 9 cm and 15 cm. If the chord of the greater circle be a tangent to the smaller circle, then the length of that chord is

Options

A. 24 cm
B. 12 cm
C. 30 cm
D. 18 cm

Answer

To find the length of the chord of the greater circle that is tangent to the smaller circle, we can follow these steps:

### Step 1: Understand the Problem
We have two concentric circles with radii 9 cm (smaller circle) and 15 cm (greater circle). A chord of the greater circle is tangent to the smaller circle.

### Step 2: Draw a Diagram
Draw two concentric circles. Mark the center as O. Label the radius of the smaller circle (9 cm) and the radius of the larger circle (15 cm). Draw a chord AB of the larger circle that is tangent to the smaller circle at point C.

### Step 3: Identify Key Points
Since the chord AB is tangent to the smaller circle at point C, the radius OC (from the center O to the point of tangency C) is perpendicular to the chord AB. This means that OC is a right angle with respect to the chord.

### Step 4: Use the Right Triangle
In triangle OAC (where A and B are the endpoints of the chord and C is the point of tangency), we can apply the Pythagorean theorem. Here:
- OA = radius of the larger circle = 15 cm
- OC = radius of the smaller circle = 9 cm
- AC = half the length of the chord (let's call this x).

### Step 5: Set Up the Equation
According to the Pythagorean theorem:
\[ OA^2 = OC^2 + AC^2 \]
Substituting the known values:
\[ 15^2 = 9^2 + x^2 \]

### Step 6: Calculate
Now, calculate the squares:
\[ 225 = 81 + x^2 \]
Subtract 81 from both sides:
\[ 225 - 81 = x^2 \]
\[ 144 = x^2 \]

### Step 7: Solve for x
Taking the square root of both sides:
\[ x = \sqrt{144} \]
\[ x = 12 \text{ cm} \]

### Step 8: Find the Length of the Chord
Since x is half the length of the chord AB, the total length of the chord is:
\[ AB = 2x = 2 \times 12 = 24 \text{ cm} \]

### Final Answer
The length of the chord is **24 cm**.

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