To solve this problem, let's break it down into key steps:

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Step 1: Understand the problem setup

1. Two circles touch externally at a single point, meaning the distance between their centers is equal to the sum of their radii. Let the radii of the two circles be $R_1$ and $R_2$.
2. The ratio of their radii is given as: $R_1 = \frac{5}{13} R_2$
3. The distance between the centers of the two circles is 36 cm: $R_1 + R_2 = 36$

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Step 2: Express $R_1$ and $R_2$ using the ratio

From the given ratio, $R_1 = \frac{5}{13} R_2$. Substituting this into the equation $R_1 + R_2 = 36$, we have: $\frac{5}{13} R_2 + R_2 = 36$

Simplify the left-hand side: $\frac{5}{13} R_2 + \frac{13}{13} R_2 = \frac{18}{13} R_2$

This gives: $\frac{18}{13} R_2 = 36$

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Step 3: Solve for $R_2$

Multiply through by 13 to eliminate the fraction: $18 R_2 = 36 \times 13$ $18 R_2 = 468$

Divide by 18: $R_2 = \frac{468}{18} = 26 \, \text{cm}$

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Step 4: Find $R_1$

Using the ratio $R_1 = \frac{5}{13} R_2$, substitute $R_2 = 26$: $R_1 = \frac{5}{13} \times 26 = 10 \, \text{cm}$

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Final Answer:

The radii of the circles are: $R_1 = 10 \, \text{cm}, \quad R_2 = 26 \, \text{cm}$

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Do you want further clarification or a diagram for this? Here are some related questions for exploration:

1. How does the condition of external tangency affect the relationship between the centers of the circles?
2. What happens if the circles touch internally instead of externally?
3. How can we generalize this solution for any ratio between $R_1$ and $R_2$?
4. Can you derive the equation if the ratio is expressed as $R_2 = k R_1$?
5. How does the distance between the centers change if $R_1 + R_2 \neq 36$?

Tip:

When dealing with geometric problems involving circles, always consider the relationship between the radii and the distance between the centers. This often provides crucial insights!