Two concentric circles are of radii 10 cm and 6 cm. Find the length of the chord of the longer circle which toiches the smaller circle.

Options

A. 10 cm
B. 12 cm
C. 15 cm
D. 16 cm

Answer

To find the length of the chord of the longer circle that touches the smaller circle, we can follow these steps:

### Step 1: Understand the Problem
We have two concentric circles. The radius of the larger circle (R) is 10 cm, and the radius of the smaller circle (r) is 6 cm. We need to find the length of the chord of the larger circle that touches the smaller circle.

### Step 2: Draw the Diagram
Draw two concentric circles. Label the center of the circles as O. Let A and B be the points where the chord touches the larger circle, and C be the point where the chord touches the smaller circle.

### Step 3: Identify the Right Triangle
Since the chord AB touches the smaller circle at point C, we can draw a radius OC from the center O to point C. This radius is perpendicular to the chord AB at point C. Thus, we have a right triangle OAC.

### Step 4: Apply the Pythagorean Theorem
In triangle OAC:
- OA (the radius of the larger circle) = 10 cm
- OC (the radius of the smaller circle) = 6 cm
- AC is half the length of the chord AB, which we will denote as x.

According to the Pythagorean theorem:
\[ OA^2 = OC^2 + AC^2 \]
Substituting the known values:
\[ 10^2 = 6^2 + x^2 \]
\[ 100 = 36 + x^2 \]

### Step 5: Solve for x
Now, rearranging the equation to find x:
\[ x^2 = 100 - 36 \]
\[ x^2 = 64 \]
\[ x = \sqrt{64} \]
\[ x = 8 \text{ cm} \]

### Step 6: Calculate the Length of the Chord
Since AC is half the length of the chord AB, the total length of the chord AB is:
\[ AB = 2 \times AC = 2 \times 8 = 16 \text{ cm} \]

### Final Answer
The length of the chord of the longer circle which touches the smaller circle is **16 cm**.

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