The radius of the two concentric circles is 5 cm and 13 cm. The tangent to the smaller circle is the chord to the greater circle. What is the length (in cm) of that chord?

Options

A. 12
B. 20
C. 22
D. 24

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:
- Radius of the smaller circle (r1) = 5 cm
- Radius of the larger circle (r2) = 13 cm

The tangent to the smaller circle is also a chord of the larger circle.

### Step 2: Identify the Points
Let:
- O be the center of both circles.
- A and B be the endpoints of the chord on the larger circle.
- M be the midpoint of chord AB.
- OM is the perpendicular distance from the center O to the chord AB.

### Step 3: Use the Properties of Tangents and Chords
Since OM is the radius of the smaller circle, we have:
- OM = r1 = 5 cm

### Step 4: Apply the Pythagorean Theorem
In triangle OAM (where OM is perpendicular to AB), we can apply the Pythagorean theorem:
\[ OA^2 = OM^2 + AM^2 \]
Where:
- OA = r2 = 13 cm (the radius of the larger circle)
- OM = 5 cm
- AM is half the length of the chord AB.

### Step 5: Substitute the Values
Substituting the known values into the equation:
\[ 13^2 = 5^2 + AM^2 \]
\[ 169 = 25 + AM^2 \]

### Step 6: Solve for AM
Rearranging the equation gives:
\[ AM^2 = 169 - 25 \]
\[ AM^2 = 144 \]
Taking the square root:
\[ AM = \sqrt{144} = 12 \text{ cm} \]

### Step 7: Calculate the Length of Chord AB
Since M is the midpoint of AB:
\[ AB = 2 \times AM = 2 \times 12 = 24 \text{ cm} \]

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

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