**Problem:**
XY is a tangent to two circles of radii 12cm and 4cm respectively. If the circles touch each other, find the area of the region bounded by the circles and tangents.

**Solution:**
To solve this problem, we need to follow these steps:

1. **Understanding the Problem**
2. **Drawing the Diagram**
3. **Finding the Distance between the Centers of the Circles**
4. **Finding the Length of the Line Segment XY**
5. **Finding the Area of the Region Bounded by the Circles and Tangents**

**Understanding the Problem:**
We are given two circles with radii 12cm and 4cm respectively. A line, XY, is a common tangent to both circles. The circles touch each other externally. We need to find the area of the region bounded by the circles and the tangent line.

**Drawing the Diagram:**
Let's draw two circles with centers A and B and radii 12cm and 4cm respectively. The circles touch each other externally at point C. The line XY is a common tangent to both circles.

[IMAGE]

**Finding the Distance between the Centers of the Circles:**
The distance between the centers of the circles is equal to the sum of their radii, as they touch each other externally.
Let's denote the distance between the centers of the circles as AB. Since AB is the sum of the radii, we have:
AB = 12cm + 4cm = 16cm

**Finding the Length of the Line Segment XY:**
The length of XY is equal to the difference between the radii of the circles, as it is perpendicular to the common tangent.
Let's denote the length of XY as CD. Since CD is the difference between the radii, we have:
CD = 12cm - 4cm = 8cm

**Finding the Area of the Region Bounded by the Circles and Tangents:**
To find the area of the region bounded by the circles and tangents, we need to subtract the areas of the two circles from the area of the rectangle formed by AB and CD.

The area of a rectangle is given by the product of its length and width. In this case, the length is AB and the width is CD.
Let's denote the area of the rectangle as A_rect. We have:
A_rect = AB * CD = 16cm * 8cm = 128cm^2

The area of a circle is given by the formula A = πr^2, where r is the radius of the circle.
Let's denote the areas of the two circles as A1 and A2. We have:
A1 = π(12cm)^2 = 144π cm^2
A2 = π(4cm)^2 = 16π cm^2

The area of the region bounded by the circles and tangents is given by the difference between the area of the rectangle and the sum of the areas of the two circles.
Let's denote the area of the region as A_region. We have:
A_region = A_rect - (A1 + A2) = 128cm^2 - (144π cm^2 + 16π cm^2) = 128cm^2 - 160π cm^2

Therefore, the area of the region bounded by the circles and tangents is 128cm