A point with x-coordinate 9 lies on the line segment joining the points (8, 1, 1) and
(10, -2, -3). Find the coordinates of the point.

Answer

To find the coordinates of the point R with an x-coordinate of 9 that lies on the line segment joining the points A(8, 1, 1) and B(10, -2, -3), we can use the section formula.

### Step-by-Step Solution:

1. **Identify the Points:**
Let the points be:
- A = (8, 1, 1)
- B = (10, -2, -3)

2. **Use the Section Formula:**
The section formula states that if a point R divides the line segment joining points A and B in the ratio k:1, then the coordinates of R can be given by:
\[ R\left(\frac{k \cdot x_2 + 1 \cdot x_1}{k + 1}, \frac{k \cdot y_2 + 1 \cdot y_1}{k + 1}, \frac{k \cdot z_2 + 1 \cdot z_1}{k + 1}\right) \]
Here, \(x_1 = 8\), \(y_1 = 1\), \(z_1 = 1\), \(x_2 = 10\), \(y_2 = -2\), \(z_2 = -3\), and we know the x-coordinate of R is 9.

3. **Set Up the Equation for x-coordinate:**
Using the x-coordinate:
\[ 9 = \frac{k \cdot 10 + 1 \cdot 8}{k + 1} \]
Simplifying this gives:
\[ 9(k + 1) = 10k + 8 \]
Expanding and rearranging:
\[ 9k + 9 = 10k + 8 \]
\[ 9 = k + 8 \]
\[ k = 1 \]

4. **Determine the Ratio:**
Since \(k = 1\), the ratio in which point R divides the line segment AB is 1:1. This means R is the midpoint of AB.

5. **Calculate the y and z Coordinates:**
Using the midpoint formula:
\[ y = \frac{y_1 + y_2}{2} = \frac{1 + (-2)}{2} = \frac{-1}{2} = -\frac{1}{2} \]
\[ z = \frac{z_1 + z_2}{2} = \frac{1 + (-3)}{2} = \frac{-2}{2} = -1 \]

6. **Final Coordinates of Point R:**
Therefore, the coordinates of point R are:
\[ R(9, -\frac{1}{2}, -1) \]

### Final Answer:
The coordinates of the point R are \( (9, -\frac{1}{2}, -1) \).