Let the point_1 = p₁ = (1,4)
and point_2 = p₂ = (-2,1)
and Point_3 = p₃ = (x,y)
The line from point_1 to point_2 is L₁ and has slope = m₁
The line from point_1 to point_3 is L₂ and has slope = m₂
m₁ = Δy/Δx = (1-4)/(-2-1) = 1
m₂ = Δy/Δx = (y-4)/(x-1)
L₁⊥L₂ ⇒⇒⇒⇒ m₁ * m₂ = -1
∴ (y-4)/(x-1) = -1 ⇒⇒⇒ (y-4)= -(x-1)
(y-4) = (1-x) ⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒ equation (1)
The distance from point_1 to point_2 is d₁
The distance from point_1 to point_3 is d₂
d =
d₁ =
d₂ =
d₁ = d₂
∴

⇒⇒ eliminating the root
∴(-2-1)²+(1-4)² = (x-1)²+(y-4)²
(x-1)²+(y-4)² = 18
from equatoin (1) y-4 = 1-x
∴(x-1)²+(1-x)² = 18 ⇒⇒⇒⇒⇒ note: (1-x)² = (x-1)²
2 (x-1)² = 18
(x-1)² = 9
x-1 =

∴ x = 4 or x = -2
∴ y = 1 or y = 7
Point_3 = (4,1) or (-2,7)
The co ordinates of P' is (-7,-2) and Q' is (-16, -8)
<u><em>Explanation</em></u>
PQ is rotated 180 degrees clockwise about P. It means <u>P and P' are the same points</u>.
According to the graph, the coordinates of P is (-7, -2) and Q is (2, 4)
When PQ is rotated 180 degrees clockwise about P, then <u>P or P' will be the mid-point of Q and Q' </u>
Suppose, the co ordinate of Q' is (x, y)
Now according to the mid-point formula, the coordinate of P or P' will be:
, which is actually at (-7, -2)
Thus.....

So, the co ordinates of P' is (-7,-2) and Q' is (-16, -8)
5 - 1.25 = 3.75
Melody’s mom pays her sister $3.75 for emptying the dishwasher.