If AO = 6, BO= 10, and the angel of OAB is 90 degrees. What is the length of CO?

Let's analyze the problem step-by-step.

We are given:
- \( AO = 6 \)
- \( BO = 10 \)
- \(\angle OAB = 90^\circ\)

### Step-by-Step Solution

1. **Understanding the Geometry**:  
   Since \(\angle OAB = 90^\circ\), triangle \(OAB\) is a right triangle with the right angle at point \(A\).

2. **Applying the Pythagorean Theorem**:  
   For right triangle \(OAB\), the Pythagorean theorem states:
   \[
   OB^2 = OA^2 + AB^2
   \]
   Plugging in the known values:
   \[
   10^2 = 6^2 + AB^2
   \]
   \[
   100 = 36 + AB^2
   \]
   \[
   AB^2 = 64
   \]
   \[
   AB = \sqrt{64} = 8
   \]

3. **Finding the Length of CO**:  
   If \(C\) is some point (not specified in the problem) that we want to find the distance from \(O\), we need additional information. If we assume that \(CO\) is the hypotenuse of another right triangle (say \(OCB\)), or another point that lies on the line \(BO\), this might change the calculation.

However, **without further information about the position of point \(C\)**, we cannot determine the exact length of \(CO\).

### Conclusion
To find the exact length of \(CO\), additional details about point \(C\) are required. If you could provide more information about where point \(C\) is located, I could give you a more precise answer.

Would you like to clarify the location of \(C\) or ask another question?

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### Related Questions:
1. If \(C\) is the midpoint of \(AB\), what would be the length of \(CO\)?
2. How would the length of \(CO\) change if point \(C\) lies on line \(BO\)?
3. What is the perimeter of triangle \(OAB\)?
4. Can you find the area of triangle \(OAB\)?
5. If point \(C\) is on the circle centered at \(O\) with radius \(OB\), what would be the coordinates of \(C\)?

**Tip**: Always ensure to provide all known details when solving geometry problems to avoid ambiguity.

Explore how to find the length of CO in triangle OAB with AO = 6, BO = 10, and angle OAB = 90 degrees. Learn the step-by-step solution using the Pythagorean theorem and properties of right triangles. Suitable for high school geometry students.

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