The problem in the image describes a right-angled triangle with sides 6 cm, 8 cm, and 10 cm. The triangle is inscribed in a circle, meaning the circle is tangent to each side of the triangle. We are asked to find the radius of this circle.

This type of circle is known as the "incircle" of the triangle, and its radius can be determined using a specific formula for a right triangle. The formula for the radius $r$ of the incircle of a right triangle with legs $a$, $b$, and hypotenuse $c$ is:

$r = \frac{a + b - c}{2}$

Given:

• $a = 6 \, \text{cm}$
• $b = 8 \, \text{cm}$
• $c = 10 \, \text{cm}$ (the hypotenuse)

Now, substitute the values into the formula:

$r = \frac{6 + 8 - 10}{2} = \frac{4}{2} = 2 \, \text{cm}$

Thus, the radius of the circle is 2 cm.

Would you like a more detailed explanation or have any additional questions?

Here are 5 related questions to deepen understanding:

1. How is the formula for the incircle of a right triangle derived?
2. How do you calculate the area of a triangle using the incircle radius?
3. What other types of triangles can have incircles, and how are their radii calculated?
4. How does the concept of an excircle differ from an incircle in a triangle?
5. How can the Pythagorean theorem be used to verify the right triangle sides?

Tip: In any right triangle, the radius of the incircle can also give insight into the area of the triangle and help in solving various geometric problems efficiently.