To find the radius of the circle in the given triangle, we can use the following geometric relationships.

Step 1: Identify the given information

• $AB = 8$ cm
• $BC = 12$ cm
• The circle is inscribed in the right triangle $ABC$ with $\angle B = 90^\circ$.

Step 2: Find the length of the hypotenuse $AC$

Using the Pythagorean theorem: $AC = \sqrt{AB^2 + BC^2} = \sqrt{8^2 + 12^2} = \sqrt{64 + 144} = \sqrt{208} = \sqrt{16 \times 13} = 4\sqrt{13} \text{ cm}$

Step 3: Use the formula for the radius $r$ of the inscribed circle

For a right triangle, the radius $r$ of the inscribed circle can be calculated using: $r = \frac{AB + BC - AC}{2}$

Substituting the known values: $r = \frac{8 + 12 - 4\sqrt{13}}{2} = \frac{20 - 4\sqrt{13}}{2} = 10 - 2\sqrt{13} \text{ cm}$

This is the radius of the inscribed circle.

Would you like any further details or have any questions?

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Here are five related questions you might find interesting:

1. How do you derive the formula for the radius of an inscribed circle in a right triangle?
2. Can the radius of the inscribed circle be expressed in decimal form?
3. What is the area of the triangle using the sides given?
4. How would you find the radius if the triangle were not a right triangle?
5. What is the relationship between the circumradius and the inradius in a right triangle?

Tip: Always verify your calculations by cross-checking with different methods to ensure accuracy.