Answer:
Step-by-step explanation:
Step 1: What are the couch's original coordinates?
- A: (-4, 2)
- N: (-4, 3)
- G: (-1, 3)
- L: (-1, 4)
- E: (-5, 4)
- S: (-5, 2)
Step 2: Rotate the couch 90° counterclockwise.
Rule: (x, y) → (-y, x)
- A': (-4, 2) → (-2, -4)
- N': (-4, 3) → (-3, -4)
- G': (-1, 3) → (-3, -1)
- L': (-1, 4) → (-4, -1)
- E': (-5, 4) → (-4, -5)
- S': (-5, 2) → (-2, -5)
Step 3: Now reflect the "new" couch over the y-axis.
Rule: (x, y) → (-x, y)
- A'': (-2, -4) → (2, -4)
- N'': (-3, -4) → (3, -4)
- G'': (-3, -1) → (3, -1)
- L'': (-4, -1) → (4, -1)
- E'': (-4, -5) → (4, -5)
- S'': (-2, -5) → (2, -5)
Step 4: Finally translate the new new" couch right 1 unit and up 5 units.
Rule: (x + 1, y + 5)
- A''': (2, -4) → (2 + 1, -4 + 5) → (3, 1)
- N''': (3, -4) → (3 + 1, -4 + 5) → (4, 1)
- G''': (3, -1) → (3 + 1, -1 + 5) → (4, 4)
- L''': (4, -1) → (4 + 1, -1 + 5) → (5, 4)
- E''': (4, -5) → (4 + 1, -5 + 5) → (5, 0)
- S''': (2, -5) → (2 + 1, -5 + 5) → (3, 0)
Step 5: Use the distance formula to show that the length of EG is the same as the length of E'''G'''.
- EG: (-5, 4)(-1, 3)
- E'''G''': (5, 0)(4, 4)
Distance between E(-5, 4) and G(-1, 3)
- x₁ = -5
- x₂ = -1
- y₁ = 4
- y₂ = 3
[tex]\large \textsf {d = $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$}\\\\\large \textsf {d = $\sqrt{(-1-(-5))^2+(3-4)^2}$}\\\\\large \textsf {d = $\sqrt{(-1+5)^2+(3-4)^2}$}\\\\\large \textsf {d = $\sqrt{4^2+(-1)^2}$}\\\\\ \large \textsf {d = $\sqrt{16+1}$}\\\\\large \textsf {d = $\sqrt{17}$}\\\\\large \textsf {d = ${4.12}$}[/tex]
Distance between E'''(5, 0) and G'''(4, 4)
[tex]\large \textsf {d = $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$}\\\\\large \textsf {d = $\sqrt{(4-5)^2+(4-0)^2}$}\\\\\large \textsf {d = $\sqrt{(-1)^2+4^2}$}\\\\\large \textsf {d = $\sqrt{1+16}$}\\\\\ \large \textsf {d = $\sqrt{17}$}\\\\ \large \textsf {d = ${4.12}$}[/tex]
This means that the length of EG is the same as the length of E'''G'''.
Hope this helps!
