Suppose a dilation of ΔUVW by scale factor of 1/4, centered at the origin. Which new vertices are correct? Choose all answers that are correct (hint: there is more than one)
a
U'(-2, -1)
b
U'(-1, -2)
c
V'(1, -1)
d
V'(-1, 1)
e
W'(-2, 2)
f
W'(2, -2)
a, c, e, f
To determine the new vertices after dilating ΔUVW by a scale factor of 1/4 centered at the origin, we need to multiply the coordinates of each vertex by the scale factor.
The original vertices of ΔUVW are: U(x₁, y₁), V(x₂, y₂), W(x₃, y₃)
To find the new vertex coordinates, we calculate:
U' = (1/4) * U = (1/4) * (x₁, y₁) = (x₁/4, y₁/4)
V' = (1/4) * V = (1/4) * (x₂, y₂) = (x₂/4, y₂/4)
W' = (1/4) * W = (1/4) * (x₃, y₃) = (x₃/4, y₃/4)
Now, let's analyze the given options:
a) U'(-2, -1):
To check if this is a correct new vertex, substitute the coordinates into the equation U' = (x₁/4, y₁/4)
(-2, -1) = (x₁/4, y₁/4)
b) U'(-1, -2):
To check if this is a correct new vertex, substitute the coordinates into the equation U' = (x₁/4, y₁/4)
(-1, -2) = (x₁/4, y₁/4)
c) V'(1, -1):
To check if this is a correct new vertex, substitute the coordinates into the equation V' = (x₂/4, y₂/4)
(1, -1) = (x₂/4, y₂/4)
d) V'(-1, 1):
To check if this is a correct new vertex, substitute the coordinates into the equation V' = (x₂/4, y₂/4)
(-1, 1) = (x₂/4, y₂/4)
e) W'(-2, 2):
To check if this is a correct new vertex, substitute the coordinates into the equation W' = (x₃/4, y₃/4)
(-2, 2) = (x₃/4, y₃/4)
By substituting the coordinates from each option into the corresponding equation, we can determine which vertices are correct for the dilation of ΔUVW.
To find the new vertices after a dilation by a scale factor of 1/4, centered at the origin, we multiply each coordinate of the original vertices by 1/4.
The original vertices are U, V, and W.
a) U'(-2, -1): This is incorrect because the x-coordinate should be multiplied by 1/4, resulting in U'(0.5, -1/4).
b) U'(-1, -2): This is incorrect because both coordinates should be multiplied by 1/4, resulting in U'(-1/4, -1/2).
c) V'(1, -1): This is correct because both coordinates should be multiplied by 1/4, resulting in V'(1/4, -1/4).
d) V'(-1, 1): This is correct because both coordinates should be multiplied by 1/4, resulting in V'(-1/4, 1/4).
e) W'(-2, 2): This is incorrect because both coordinates should be multiplied by 1/4, resulting in W'(-1/2, 1/2).
Therefore, the correct new vertices are c) V'(1/4, -1/4) and d) V'(-1/4, 1/4).