To find the coordinates of the new vertices of triangle X′Y′Z′ after dilating triangle XYZ, we need to use the formula for dilation given a center of dilation.
The formula for finding the coordinates \( (x', y') \) of a point \( (x, y) \) after dilation with a center of dilation at \( (a, b) \) and a magnification factor \( k \) is:
\[ x' = a + k(x - a) \] \[ y' = b + k(y - b) \]
In this case, the center of dilation is \( (4, 7) \) and the magnitude of dilation is \( 12 \).
Let's calculate the new coordinates for each vertex one by one.
For Vertex X(0, 4):
- \( a = 4 \), \( b = 7 \), \( k = 12 \)
- \( x' = 4 + 12(0 - 4) = 4 + 12(-4) = 4 - 48 = -44 \)
- \( y' = 7 + 12(4 - 7) = 7 + 12(-3) = 7 - 36 = -29 \)
So, \( X'(-44, -29) \).
For Vertex Y(2, 0):
- \( x' = 4 + 12(2 - 4) = 4 + 12(-2) = 4 - 24 = -20 \)
- \( y' = 7 + 12(0 - 7) = 7 + 12(-7) = 7 - 84 = -77 \)
So, \( Y'(-20, -77) \).
For Vertex Z(4, 7):
Since this vertex is the center of dilation, the coordinates will not change.
- \( Z' = (4, 7) \).
Putting it all together:
- \( X'(-44, -29) \)
- \( Y'(-20, -77) \)
- \( Z'(4, 7) \)
However, based on the given responses, none of them correctly reflect the calculated results. It seems there may have been an error in the response choices provided or perhaps a misunderstanding in the magnitude or center of dilation. Please double-check the given values or the options. If you provide the correct values/options or clarify the parameters, I’d be glad to help re-evaluate.