To find the radius of the circle passing through the points \((15, 9)\) and \((17, 6)\) if these points are on the circumference and the center of the circle is given, we need to follow these steps:

1. **Calculate the distance between the center and any point on the circumference**. This distance will be the radius.

Let's assume the center of the circle is given by \((h, k)\). Since the center is not provided in your initial question, we will need to find the distance between the two points themselves to calculate their midpoint, which could potentially be the center if we have no other reference. If \((x_1, y_1)\) and \((x_2, y_2)\) are the given points, the distance (which is also the diameter) is calculated using the distance formula:

\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

2. **Divide the distance by 2 to find the radius**.

Given:
\((x_1, y_1) = (15, 9)\)
\((x_2, y_2) = (17, 6)\)

Calculate the distance between these points:

\[ \text{Distance} = \sqrt{(17 - 15)^2 + (6 - 9)^2} \]
\[ \text{Distance} = \sqrt{2^2 + (-3)^2} \]
\[ \text{Distance} = \sqrt{4 + 9} \]
\[ \text{Distance} = \sqrt{13} \]

Since the distance calculated is the diameter of the circle, the radius \(r\) is half of the diameter:

\[ r = \frac{\sqrt{13}}{2} \]

Thus, the radius of the circle is:

\[ r = \frac{\sqrt{13}}{2} \approx 1.8 \]

Would you like more details or have any questions?

Here are 8 related questions to expand on this topic:

1. How do you find the equation of a circle given its center and radius?
2. What is the length of the diameter of the circle if the radius is \(\frac{\sqrt{13}}{2}\)?
3. How would you find the midpoint between two given points?
4. Can you derive the general form of a circle's equation from its standard form?
5. How can you determine if a given point lies inside, on, or outside the circle?
6. What is the area of a circle with a radius of \(\frac{\sqrt{13}}{2}\)?
7. How do you find the circumference of a circle with a given radius?
8. How can you verify that two points lie on the circumference of a given circle?

**Tip:** When dealing with distances in geometry, always keep track of your units and ensure your calculations are precise to avoid errors.

Learn how to find the radius of a circle passing through two given points (15, 9) and (17, 6) using the distance formula and circle geometry principles. This problem involves calculating the distance between points, determining the radius from the distance, and applying circle geometry concepts. Suitable for high school students.

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