Problem: Two concentric circles are of radii 7 cm and r respectively, where r > 7. A chord of the larger circle, with a length of 48 cm, touches the smaller circle. Find the value of r.

Solution:

To find the value of r, we need to use the properties of tangents and chords in circles.

Step 1: Draw a diagram
Draw two concentric circles, one with a radius of 7 cm and the other with a radius of r. Draw a chord of the larger circle that touches the smaller circle.

Step 2: Identify the given information
We are given that the length of the chord is 48 cm and that it touches the smaller circle.

Step 3: Identify the key relationships
There are a few key relationships that we can use to solve this problem:

- A tangent to a circle is perpendicular to the radius at the point of contact.
- If a chord in a circle is perpendicular to a radius, it bisects the radius.
- The length of a chord that is perpendicular to a radius is equal to twice the distance from the center of the circle to the chord.

Step 4: Apply the key relationships
Since the chord of the larger circle touches the smaller circle, it is perpendicular to the radius at the point of contact. Therefore, it bisects the radius of the larger circle.

Let O be the center of the concentric circles, and let M be the point where the chord touches the smaller circle. The line segment OM is the radius of the larger circle, and it is bisected by the chord.

Let the length of the chord be x. According to the key relationships, we have:

OM = 2 * x

Since the length of the chord is given as 48 cm, we can rewrite the equation as:

2 * x = 48

Step 5: Solve the equation
Solving the equation, we find:

x = 24

Since the chord bisects the radius of the larger circle, we can now write the equation:

OM = r - 7

Substituting the value of x, we have:

24 = r - 7

Solving for r, we find:

r = 24 + 7 = 31

Therefore, the value of r is 31 cm.