The radius of two concentric circles are 17cm and 10cm. A straight line ABCD intersects the larger circle at the point A and D and intersects the smaller circle at the points B and C. If BC = 12 cm, then the length of AD (in cm) is :

Options

A. 20
B. 24
C. 30
D. 34

Answer

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

### Step 1: Understand the Problem
We have two concentric circles with radii 17 cm (larger circle) and 10 cm (smaller circle). A straight line ABCD intersects the larger circle at points A and D, and the smaller circle at points B and C. We know that BC = 12 cm.

### Step 2: Identify Key Points
Let O be the center of the circles. The line segment BC is perpendicular to the radius at point O, and it bisects the segment BC. Therefore, we can find the lengths of BE and CE:
- BE = CE = BC / 2 = 12 cm / 2 = 6 cm.

### Step 3: Apply the Pythagorean Theorem
We can use the Pythagorean theorem in two right triangles: OBE and OAE.

1. For triangle OBE:
- OB (radius of smaller circle) = 10 cm
- BE = 6 cm
- OE (the perpendicular from O to BC) can be calculated as:
\[ OE^2 + BE^2 = OB^2 \\ OE^2 + 6^2 = 10^2 \\ OE^2 + 36 = 100 \\ OE^2 = 64 \\ OE = 8 \text{ cm} \]

2. For triangle OAE:
- OA (radius of larger circle) = 17 cm
- AE (the segment we want to find) can be calculated as:
\[ OE^2 + AE^2 = OA^2 \\ 8^2 + AE^2 = 17^2 \\ 64 + AE^2 = 289 \\ AE^2 = 289 - 64 \\ AE^2 = 225 \\ AE = 15 \text{ cm} \]

### Step 4: Calculate Length of AD
Since AD is a chord of the larger circle, and OE bisects AD, we have:
- AE = ED = 15 cm
- Therefore, the total length of AD is:
\[ AD = AE + ED = 15 + 15 = 30 \text{ cm} \]

### Final Answer
The length of AD is **30 cm**.

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