the segment joining (-1,4) and (-2,2) is extended three times its own length. find the point nearer to (-1,4)

To find the point that is nearer to (-1,4) when a segment is extended three times its own length, we can follow these steps:

1. Find the length of the segment joining (-1,4) and (-2,2).
- The distance formula is used to find the distance between two points in a plane.
- The length of the segment is calculated by applying the distance formula to the given points.
- Using the distance formula:
- d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
- d = sqrt((-2 - (-1))^2 + (2 - 4)^2)
- d = sqrt((-2 + 1)^2 + (2 - 4)^2)
- d = sqrt((1)^2 + (-2)^2)
- d = sqrt(1 + 4)
- d = sqrt(5)

2. Calculate three times the length of the segment.
- Multiply the length of the segment by 3.
- 3 * sqrt(5)

3. Extend the segment by three times its own length from the initial point (-1,4).
- Starting from the initial point (-1,4), move in the direction of the point (-2,2) and extend the segment three times its length.
- Determine the new coordinates by multiplying the direction vector of the segment (-1 - (-2), 4 - 2) by three times the length.
- The extended segment can be represented by the equation:
- x = -1 + 3 * (x2 - x1)
- y = 4 + 3 * (y2 - y1)
- x = -1 + 3 * (-2 - (-1))
- y = 4 + 3 * (2 - 4)
- x = -1 + 3 * (-2 + 1)
- y = 4 + 3 * (2 - 4)
- x = -1 + 3 * (-1)
- y = 4 + 3 * (-2)
- x = -1 + (-3)
- y = 4 + (-6)
- x = -4
- y = -2

4. Compare the distances of the two points (-2,2) and (-4,-2) from the initial point (-1,4).
- Calculate the distance between the initial point (-1,4) and both points (-2,2) and (-4,-2) using the distance formula.
- Distance between (-1,4) and (-2,2):
- d1 = sqrt((-2 - (-1))^2 + (2 - 4)^2)
- d1 = sqrt((-2 + 1)^2 + (2 - 4)^2)
- d1 = sqrt((1)^2 + (-2)^2)
- d1 = sqrt(1 + 4)
- d1 = sqrt(5)
- Distance between (-1,4) and (-4,-2):
- d2 = sqrt((-4 - (-1))^2 + (-2 - 4)^2)
- d2 = sqrt((-4 + 1)^2 + (-2 - 4)^2)
- d2 = sqrt((-3)^2 + (-6)^2)
- d2 = sqrt(9 + 36)
- d2 = sqrt(45)
- d2 = sqrt(9 * 5)
- d2 = 3 * sqrt(5)

- Compare the values of d1 and d2.
- d1 = sqrt(5)
- d2 = 3 * sqrt(5)

As d2 (3 * sqrt(5)) is greater than d1 (sqrt(5)), the point (-4,-2) is nearer to (-1,4) than (-2,2).