The problem involves finding the length of the segment $LC$ in terms of $R$, given two tangent circles of radius $R$ that are touching both each other and a horizontal line.

Problem Analysis:

1. The problem has two circles, each with radius $R$.
2. The leftmost circle has its center at $L$ and the rightmost circle has its center at $C$.
3. The distance between the centers of the two circles is $2R$, since they are tangent.

Step-by-step Solution:

1. The segment $LC$ can be broken down into:

   • The horizontal distance from $L$ to $C$, which is $2R$.
   • The vertical component due to the triangle formed by the points $L$, $C$, and the point of tangency.

2. Let’s now solve geometrically by using the properties of the triangle.

I'll now proceed to analyze and calculate the exact expression for $LC$.The length of the segment $LC$ in terms of $R$ is:

$LC = R(\sqrt{2} + 2)$

Now let's compare this with the multiple-choice options provided in the image. Unfortunately, the expression $R(\sqrt{2} + 2)$ does not directly match any of the available choices. Let's examine each option more closely:

• Option A: $R\sqrt{2} + \sqrt{3}$
• Option B: $R\sqrt{3} + \sqrt{2}$
• Option C: $R\sqrt{2} + \sqrt{3}$
• Option D: $R(\sqrt{2} - \sqrt{3})$
• Option E: $R(\sqrt{3} - \sqrt{2})$

Given this, it appears none of the options exactly represent $R(\sqrt{2} + 2)$, suggesting either a mistake in the problem or a misinterpretation of the geometric setup.

Would you like further details or an alternative analysis of the problem?

Related Questions:

1. How can we derive the distance between two tangent circles?
2. What role does the Pythagorean theorem play in this geometric problem?
3. How do properties of regular polygons relate to this problem's solution?
4. How can we express the sum of distances in tangent circles geometrically?
5. Can alternative methods like coordinate geometry simplify such calculations?

Tip:

In problems involving tangent circles, always check if the distance between their centers matches the sum of their radii.