The line segment joining P1 (0,-5) and P2 (4,1) is extended beyond P1 to Pr so that Pr is 4 times as far from P2 as from P1. Find the coordinates of Pr
Let the coordinates of Pr = (x, y)
So we have
Four times the distance from the x coordinate of Pr to the x coordinate of P1 = the distance from the x coordinate of Pr to the x coordinate of P2.......expressed mathematically, we have
4(x - 0) = x - 4
4x = x - 4
3x = -4
x = -4/3
And four times the distance from the y coordinate of Pr to the y coordinate of P1 = the distance from the y coordinate of Pr to the y coordinate of P2.......expressed mathematically.......
4(y-(-5)) = y - 1
4(y+5) = y - 1
4y + 20 = y - 1
3y = -21
y = -7
So Pr = (-4/3, -7)
Proof
Distance from Pr to P1
√[(-4/3 - 0)^2 + (-7- (-5))^2] = √[(-4/3)^2 + (-2)^2] = about 2.4037
Distance from Pr To P2
√[(-4/3 - 4)^2 + (-7- 1)^2] = √[(-16/3 )^2 + (-8)^2] = about 9.6148
And 9.6148034012373047793 / 4 = 2.4037
Let the coordinates of Pr = (x, y)
So we have
Four times the distance from the x coordinate of Pr to the x coordinate of P1 = the distance from the x coordinate of Pr to the x coordinate of P2.......expressed mathematically, we have
4(x - 0) = x - 4
4x = x - 4
3x = -4
x = -4/3
And four times the distance from the y coordinate of Pr to the y coordinate of P1 = the distance from the y coordinate of Pr to the y coordinate of P2.......expressed mathematically.......
4(y-(-5)) = y - 1
4(y+5) = y - 1
4y + 20 = y - 1
3y = -21
y = -7
So Pr = (-4/3, -7)
Proof
Distance from Pr to P1
√[(-4/3 - 0)^2 + (-7- (-5))^2] = √[(-4/3)^2 + (-2)^2] = about 2.4037
Distance from Pr To P2
√[(-4/3 - 4)^2 + (-7- 1)^2] = √[(-16/3 )^2 + (-8)^2] = about 9.6148
And 9.6148034012373047793 / 4 = 2.4037