To find the radius of circle $O$, we can use the properties of the circle and the given lengths of the triangle $\triangle ABC$.

Here are the steps:

1. Recognize the Inscribed Angle Theorem:

   • Since $AB$ is a diameter of the circle, $\angle ACB$ is a right angle (90 degrees). This is due to the inscribed angle theorem, which states that an angle inscribed in a semicircle is a right angle.

2. Use the Pythagorean Theorem:

   • Given $\angle ACB$ is 90 degrees, $\triangle ACB$ is a right triangle with hypotenuse $AB$, and legs $AC$ and $BC$.
   • We are given $AC = 12$ and $BC = 5$.

3. Calculate the Diameter $AB$:

   • Using the Pythagorean theorem: $AB^2 = AC^2 + BC^2$ $AB^2 = 12^2 + 5^2$ $AB^2 = 144 + 25$ $AB^2 = 169$ $AB = \sqrt{169} = 13$

4. Determine the Radius:

   • The diameter $AB$ is 13 units. Therefore, the radius $O$ of the circle is half of the diameter: $\text{Radius} = \frac{AB}{2} = \frac{13}{2} = 6.5 \text{ units}$

Thus, the radius of circle $O$ is $6.5$ units.

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How do you find the radius of a circle if only one chord length is given?
2. Can you explain the inscribed angle theorem in more detail?
3. How would the problem change if the given lengths were different?
4. What other methods can be used to determine the radius of a circle?
5. How do the properties of circles apply to real-world problems?

Tip: Always check if a triangle is a right triangle when dealing with circle diameters, as it can simplify your calculations significantly.