You are just beginning your life as an adult and have bought a house in a new housing development. Your house is located at A (2,3), B (4,3), C (4,5), D (3,6), and E (2,5). A flood moves your house exactly 4 blocks to the left and 2 blocks down. Is your new house after this transformation congruent to the original house? Explain your answer.
To determine if the new house is congruent to the original house, we need to compare the distances and angles between the corresponding points of the two houses.
Original House:
A(2,3)
B(4,3)
C(4,5)
D(3,6)
E(2,5)
Transformed House:
A'(-2,1)
B'(-4,1)
C'(-4,3)
D'(-3,4)
E'(-2,3)
Comparing the distances:
AB = 2 units
A'B' = 2 units
BC = √((4-4)^2 + (5-3)^2) = √(0+4) = 2 units
B'C' = √((-4+4)^2 + (3-1)^2) = √(0+4) = 2 units
CD = √((4-3)^2 + (5-6)^2) = √(1+1) = √2 units
C'D' = √((-4+3)^2 + (3-4)^2) = √(1+1) = √2 units
DE = 1 unit
D'E' = 1 unit
EA = √((2-2)^2 + (3-5)^2) = √(0+4) = 2 units
E'A' = √((-2+2)^2 + (1-3)^2) = √(0+4) = 2 units
Comparing the angles:
The angle between AB and BC is 90° in both the original house and transformed house.
The angle between BC and CD is approximately 135° in both the original house and transformed house.
The angle between CD and DE is approximately 45° in both the original house and transformed house.
The angle between DE and EA is approximately 135° in both the original house and transformed house.
Since all the distances and angles between the corresponding points of the original house and the transformed house are equal, we can conclude that the two houses are congruent.