The radii of two concentric circle are 13 cm and 8 cm. AB is a diameter of the bigger circle and BD is a tangent to the smaller circle touching it at D and the bigger circle at E. Point A is joined to D. Find the length of AD.

Options

A. 20 cm
B. 19 cm
C. 18 cm
D. 17 cm

Answer

To find the length of AD in the given problem, we can follow these steps:

### Step 1: Understand the Geometry
We have two concentric circles with radii 13 cm (larger circle) and 8 cm (smaller circle). The diameter AB of the larger circle is 26 cm (since diameter = 2 * radius). The tangent BD touches the smaller circle at point D and the larger circle at point E.

### Step 2: Draw the Diagram
Draw the two concentric circles, label the center O, and mark points A and B on the larger circle. Draw the tangent line BD, touching the smaller circle at point D and the larger circle at point E.

### Step 3: Identify Right Angles
Since BD is a tangent to the smaller circle at point D, it is perpendicular to the radius OD at point D. Thus, we have:
- Angle ODB = 90 degrees.

### Step 4: Apply the Pythagorean Theorem
In triangle OBD, we can apply the Pythagorean theorem:
\[ OB^2 = OD^2 + BD^2 \]

Where:
- \( OB = 13 \, \text{cm} \) (radius of the larger circle),
- \( OD = 8 \, \text{cm} \) (radius of the smaller circle).

Substituting the values:
\[ 13^2 = 8^2 + BD^2 \]
\[ 169 = 64 + BD^2 \]
\[ BD^2 = 169 - 64 \]
\[ BD^2 = 105 \]

### Step 5: Find Length of AE
Since D is the midpoint of BE (because the radius bisects the chord), we can find AE:
- \( AE = 2 \times OD = 2 \times 8 = 16 \, \text{cm} \).

### Step 6: Apply Pythagorean Theorem in Triangle ADE
Now, in triangle ADE, we can again apply the Pythagorean theorem:
\[ AD^2 = DE^2 + AE^2 \]

Since \( DE = BD \) (because BD is tangent to both circles), we have:
- \( DE^2 = BD^2 = 105 \),
- \( AE^2 = 16^2 = 256 \).

Substituting these values:
\[ AD^2 = 105 + 256 \]
\[ AD^2 = 361 \]

### Step 7: Calculate AD
Taking the square root to find AD:
\[ AD = \sqrt{361} = 19 \, \text{cm} \].

Thus, the length of AD is **19 cm**.

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