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Write the coordinates of any two points on the line segment joining the points [tex]\(A(4, -1)\)[/tex] and [tex]\(B(4, 5)\)[/tex].

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GinnyAnsweridn
Sure! Let's determine the coordinates of two points on the line segment joining points A (4, -1) and B (4, 5).

### Step-by-Step Solution:

1. Determine the Midpoint of Segment AB:
To find the midpoint between two points, we use the midpoint formula:
\( \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)

For our points A (4, -1) and B (4, 5):
- The x-coordinate of the midpoint is \( \frac{4 + 4}{2} = 4.0 \)
- The y-coordinate of the midpoint is \( \frac{-1 + 5}{2} = 2.0 \)

So, the coordinates of the midpoint are \( (4.0, 2.0) \).

2. Determine a Point Located One-Third of the Distance from A to B:
To find a point that divides the segment in a given ratio, say \( m:n \), we use the section formula:
\( P = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \)

For a point one-third of the way from A to B (or dividing the segment in the ratio 1:2):
- The x-coordinate remains 4 because the x-coordinates of both points are the same.
- The y-coordinate is given by \( \frac{1 \cdot 5 + 2 \cdot (-1)}{1 + 2} = \frac{5 - 2}{3} = 1.0 \)

Thus, the coordinates of the point one-third of the distance from A to B are \( (4, 1.0) \).

### Final Coordinates:
- Midpoint: \( (4.0, 2.0) \)
- One-third Point: \( (4, 1.0) \)

These are the coordinates of two points on the line segment joining A (4, -1) and B (4, 5).
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