Two concentric circles are of radii 5cm and 3cm. Find the length of chord of the larger circle which
touches the smaller circle.

To find the length of the chord of the larger circle that touches the smaller circle, we can use geometry to set up the problem.

### Key Information:
- The two circles are concentric, meaning they share the same center.
- The radius of the larger circle is \( R = 5 \, \text{cm} \).
- The radius of the smaller circle is \( r = 3 \, \text{cm} \).
- The chord of the larger circle touches the smaller circle. This means the distance from the center to the chord is equal to the radius of the smaller circle, \( r = 3 \, \text{cm} \).

### Step-by-Step Solution:

1. **Draw a diagram**: Consider the center of the circles as \( O \), the midpoint of the chord as \( P \), and the points where the chord intersects the larger circle as \( A \) and \( B \).

2. **Use a right triangle**: 
   - The line segment \( OP \) (from the center \( O \) to the midpoint \( P \) of the chord) is perpendicular to the chord and has a length equal to the radius of the smaller circle, i.e., \( OP = 3 \, \text{cm} \).
   - The radius of the larger circle is \( OA = OB = 5 \, \text{cm} \).
   - Triangle \( OAP \) is a right triangle where \( OA = 5 \, \text{cm} \), \( OP = 3 \, \text{cm} \), and we need to find \( AP \), which is half of the length of the chord.

3. **Apply the Pythagorean theorem**:
   In the right triangle \( OAP \):
   \[
   OA^2 = OP^2 + AP^2
   \]
   Substituting the values:
   \[
   5^2 = 3^2 + AP^2
   \]
   \[
   25 = 9 + AP^2
   \]
   \[
   AP^2 = 25 - 9 = 16
   \]
   \[
   AP = \sqrt{16} = 4 \, \text{cm}
   \]

4. **Find the length of the chord**:
   Since \( AP = 4 \, \text{cm} \), the full length of the chord \( AB \) is:
   \[
   AB = 2 \times AP = 2 \times 4 = 8 \, \text{cm}
   \]

### Final Answer:
The length of the chord of the larger circle that touches the smaller circle is \( 8 \, \text{cm} \).

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Would you like further details on this, or do you have any questions?

### Related Questions:
1. What is the length of a chord in a circle if the perpendicular distance from the center is given?
2. How do you find the distance between two concentric circles?
3. What is the formula for the length of a chord at a given distance from the center?
4. How do you derive the equation of a circle from its radius and center?
5. How would the problem change if the two circles were not concentric?

### Tip:
In problems involving chords and circles, using the Pythagorean theorem in conjunction with radii and perpendiculars to the chord is a common and effective approach!

Two concentric circles are of radii 5cm and 3cm. Find the length of chord of the larger circle which touches the smaller circle.

Learn how to calculate the length of a chord in a larger circle that touches a smaller concentric circle. The solution uses geometry and the Pythagorean theorem. This problem is ideal for students in Grades 9-10 and covers important concepts related to concentric circles and chord properties.

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