Question
If you dilate triangle XYZ to form X′Y′Z′ using a magnitude of 2 and a center of dilation at (2,0) , what are the coordinates of the dilated triangle? The coordinates are, X: (2,4) Y:6, 0), And Z: (7, 2) (1 point) Responses X′(6,4) , Y′(10,0) , Z′(11,4) upper X prime left parenthesis 4 comma 8 right parenthesis , upper Y prime left parenthesis 12 comma 0 right parenthesis , upper Z prime left parenthesis 14 comma 4 right parenthesis X′(2,6) , Y′(8,0) , Z′(9.5,3) upper X prime left parenthesis 8 comma 4 right parenthesis , upper Y prime left parenthesis 12 comma 0 right parenthesis , upper Z prime left parenthesis 13 comma 2 right parenthesis X′(2,8) , Y′(10,0) , Z′(12,4) upper X prime left parenthesis negative 2 comma 8 right parenthesis , upper Y prime left parenthesis 6 comma 0 right parenthesis , upper Z prime left parenthesis 8 comma 4 right parenthesis X′(2,2) , Y′(4,0) , Z′(4.5,1) upper X prime left parenthesis negative 4 comma 4 right parenthesis , upper Y prime left parenthesis 0 comma 0 right parenthesis , upper Z prime left parenthesis 1 comma 2 right parenthesis
Answers
GPT-4o mini
To find the coordinates of the dilated triangle \( X'Y'Z' \) from triangle \( XYZ \) using a center of dilation at \( (2,0) \) and a magnitude of \( 2 \), we can use the dilation formula.
The formula for dilation with a center \( (x_c, y_c) \) and a scale factor \( k \) for a point \( (x, y) \) is given by:
\[
(x', y') = (x_c + k(x - x_c), y_c + k(y - y_c))
\]
In this case, \( (x_c, y_c) = (2,0) \) and \( k = 2 \).
We will apply this to each vertex of triangle \( XYZ \):
1. **For point \( X(2, 4) \)**:
\[
X' = \left(2 + 2(2 - 2), 0 + 2(4 - 0)\right) = \left(2 + 0, 0 + 8\right) = (2, 8)
\]
2. **For point \( Y(6, 0) \)**:
\[
Y' = \left(2 + 2(6 - 2), 0 + 2(0 - 0)\right) = \left(2 + 8, 0 + 0\right) = (10, 0)
\]
3. **For point \( Z(7, 2) \)**:
\[
Z' = \left(2 + 2(7 - 2), 0 + 2(2 - 0)\right) = \left(2 + 10, 0 + 4\right) = (12, 4)
\]
The coordinates of the dilated triangle \( X'Y'Z' \) are:
- \( X' (2, 8) \)
- \( Y' (10, 0) \)
- \( Z' (12, 4) \)
From the options provided, the correct answer appears to reflect the dilated coordinates as \( X' (2, 8), Y' (10, 0), Z' (12, 4) \). Please verify the options to see if they match these values.
The formula for dilation with a center \( (x_c, y_c) \) and a scale factor \( k \) for a point \( (x, y) \) is given by:
\[
(x', y') = (x_c + k(x - x_c), y_c + k(y - y_c))
\]
In this case, \( (x_c, y_c) = (2,0) \) and \( k = 2 \).
We will apply this to each vertex of triangle \( XYZ \):
1. **For point \( X(2, 4) \)**:
\[
X' = \left(2 + 2(2 - 2), 0 + 2(4 - 0)\right) = \left(2 + 0, 0 + 8\right) = (2, 8)
\]
2. **For point \( Y(6, 0) \)**:
\[
Y' = \left(2 + 2(6 - 2), 0 + 2(0 - 0)\right) = \left(2 + 8, 0 + 0\right) = (10, 0)
\]
3. **For point \( Z(7, 2) \)**:
\[
Z' = \left(2 + 2(7 - 2), 0 + 2(2 - 0)\right) = \left(2 + 10, 0 + 4\right) = (12, 4)
\]
The coordinates of the dilated triangle \( X'Y'Z' \) are:
- \( X' (2, 8) \)
- \( Y' (10, 0) \)
- \( Z' (12, 4) \)
From the options provided, the correct answer appears to reflect the dilated coordinates as \( X' (2, 8), Y' (10, 0), Z' (12, 4) \). Please verify the options to see if they match these values.