The segment joining (-1,4), (2,-2) is extended three times its own length. Find the terminal points.
To find the terminal points of the segment joining (-1,4) and (2,-2) extended three times its own length, we need to find the new endpoint coordinates.
1. Calculate the length of the given segment:
- Length = √((x2 - x1)^2 + (y2 - y1)^2)
- Length = √((2 - (-1))^2 + (-2 - 4)^2)
- Length = √((3)^2 + (-6)^2)
- Length = √(9 + 36)
- Length = √45
- Length ≈ 6.71
2. Extend the segment three times its length:
- Extended Length = 3 * Length
- Extended Length = 3 * 6.71
- Extended Length ≈ 20.13
3. Calculate the coordinates of the new endpoint:
- To extend the segment three times its length, we can use the formula:
- x3 = x1 + (x2 - x1) * (Extended Length / Length)
- y3 = y1 + (y2 - y1) * (Extended Length / Length)
- Substituting the given values:
- x3 = -1 + (2 - (-1)) * (20.13 / 6.71)
- y3 = 4 + (-2 - 4) * (20.13 / 6.71)
- Calculating:
- x3 = -1 + 3 * (20.13 / 6.71)
- x3 ≈ -1 + 3 * 3
- x3 ≈ -1 + 9
- x3 ≈ 8
- y3 = 4 + (-6) * (20.13 / 6.71)
- y3 ≈ 4 + (-6) * 3
- y3 ≈ 4 - 18
- y3 ≈ -14
Therefore, the terminal points of the segment joining (-1,4) and (2,-2) extended three times its own length are (8, -14).
To find the terminal points of a segment extended three times its own length, we need to perform a few steps:
Step 1: Find the length of the given segment.
Using the distance formula, we can calculate the distance between the two given points (-1, 4) and (2, -2).
The formula for finding the distance between two points (x1, y1) and (x2, y2) is:
distance = √((x2 - x1)^2 + (y2 - y1)^2)
Applying this formula to the given points:
distance = √((2 - (-1))^2 + (-2 - 4)^2)
= √((3)^2 + (-6)^2)
= √(9 + 36)
= √45
= 3√5
So, the length of the given segment is 3√5.
Step 2: Extend the segment to three times its length.
To extend the segment, multiply its length by 3:
extended length = 3 * 3√5
= 9√5
Step 3: Determine the terminal points.
Since the segment is extended in both directions, we need to find the terminal points by adding and subtracting the extended length from each coordinate of the given points.
For the first given point (-1, 4):
Terminal point 1 = (-1, 4) + (9√5, 9√5)
= (-1 + 9√5, 4 + 9√5)
For the second given point (2, -2):
Terminal point 2 = (2, -2) - (9√5, 9√5)
= (2 - 9√5, -2 - 9√5)
Therefore, the terminal points of the segment extended three times its own length are:
Terminal point 1 = (-1 + 9√5, 4 + 9√5)
Terminal point 2 = (2 - 9√5, -2 - 9√5)
let the end point be (x,y)
for the x-coordinate:
(x+1)/(2+1) = 3/1
x+1 =9
x = 8
for the y-coordinate:
(y-4)/(-2-4) = 3/1
y-4 = -18
y = -14
one end point is (8,-14)
we could have extended it the other way ...
(x-2)(-1-2) = 3/1
x-2 = -9
x= -7
(y+2)/(4+2) = 3/1
y+2 = 18
y = 16
in that direction the point is (-7,16)
check my arithmetic.