Two concentric circles area of radii 8cm and 5cm. Find the length of the chord of the length circle which touches the smaller circle.

Answer

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

### Step 1: Understand the Problem
We have two concentric circles with radii 8 cm (larger circle) and 5 cm (smaller circle). We need to find the length of a chord of the larger circle that touches the smaller circle.

**Hint:** Visualize the problem by drawing two concentric circles and labeling their centers and radii.

### Step 2: Label the Points
Let:
- O be the center of both circles.
- A and B be the endpoints of the chord on the larger circle.
- C be the point where the chord touches the smaller circle.

**Hint:** Remember that the chord AB will be perpendicular to the radius OC at point C.

### Step 3: Apply the Right Triangle Concept
Since the chord touches the smaller circle at point C, we know that the radius OC (5 cm) is perpendicular to the chord AB at point C. Therefore, triangle OAC is a right triangle.

**Hint:** Use the property that the radius to the point of tangency is perpendicular to the tangent line (the chord in this case).

### Step 4: Use the Pythagorean Theorem
In triangle OAC:
- OA = 8 cm (radius of the larger circle)
- OC = 5 cm (radius of the smaller circle)
- AC = ? (half the length of the chord)

Using the Pythagorean theorem:
\[ OA^2 = OC^2 + AC^2 \]
\[ 8^2 = 5^2 + AC^2 \]
\[ 64 = 25 + AC^2 \]

**Hint:** Rearranging the equation will help you isolate AC.

### Step 5: Solve for AC
Now, rearranging the equation:
\[ AC^2 = 64 - 25 \]
\[ AC^2 = 39 \]
\[ AC = \sqrt{39} \]

**Hint:** Remember that AC is only half the length of the chord.

### Step 6: Find the Length of the Chord AB
Since AB = AC + BC and AC = BC (because of symmetry), we have:
\[ AB = AC + AC = 2 \times AC \]
\[ AB = 2 \times \sqrt{39} \]

**Hint:** This step involves doubling the length of AC to find the total length of the chord.

### Final Answer
The length of the chord AB is:
\[ AB = 2\sqrt{39} \text{ cm} \]

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