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Triangle XYZ is drawn on a coordinate plane with vertices X(0,4) , Y(2,0) , and Z(4,7) . If you dilate the triangle to form tri...Question
Triangle XYZ is drawn on a coordinate plane with vertices X(0,4) , Y(2,0) , and Z(4,7) . If you dilate the triangle to form triangle X′Y′Z′ using a magnitude of 12 and a center of dilation at (4,7) , what are the coordinate points of the new vertices?(1 point) Responses X′(2,5.5) , Y′(3,3.5) , Z′(4,7) upper X prime left parenthesis 2 comma 5.5 right parenthesis , upper Y prime left parenthesis 3 comma 3.5 right parenthesis , upper Z prime left parenthesis 4 comma 7 right parenthesis X′(−4,1) , Y′(0,−7) , Z′(4,7) upper X prime left parenthesis negative 4 comma 1 right parenthesis , upper Y prime left parenthesis 0 comma negative 7 right parenthesis , upper Z prime left parenthesis 4 comma 7 right parenthesis X′(4,11) , Y′(6,7) , Z′(8,14) upper X prime left parenthesis negative 4 comma 11 right parenthesis , upper Y prime left parenthesis 6 comma 7 right parenthesis , upper Z prime left parenthesis 8 comma 14 right parenthesis X′(−4,−3) , Y′(−2,−7) , Z′(0,0)
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GPT-4o mini
To find the new coordinates of triangle \( X'Y'Z' \) after dilation, we follow these steps:
1. **Identify the center of dilation**: The center is at point \( C(4, 7) \).
2. **Calculate the scale factor**: The scale factor is given as 12.
3. **Apply the dilation formula**: The formula to find the image point \( P' \) of point \( P(x, y) \) under dilation about a center \( C(a, b) \) with a scale factor \( k \) is given by:
\[
P' = \left( a + k(x - a), b + k(y - b) \right)
\]
4. **Apply the dilation to each vertex**:
- **Vertex \( X(0, 4) \)**:
\[
X' = \left( 4 + 12(0 - 4), 7 + 12(4 - 7) \right) = \left( 4 - 48, 7 - 36 \right) = (-44, -29)
\]
- **Vertex \( Y(2, 0) \)**:
\[
Y' = \left( 4 + 12(2 - 4), 7 + 12(0 - 7) \right) = \left( 4 - 24, 7 - 84 \right) = (-20, -77)
\]
- **Vertex \( Z(4, 7) \)**:
\[
Z' = \left( 4 + 12(4 - 4), 7 + 12(7 - 7) \right) = \left( 4 + 0, 7 + 0 \right) = (4, 7)
\]
5. **New Vertices**: The coordinates of the new vertices after dilation are:
- \( X'(-44, -29) \)
- \( Y'(-20, -77) \)
- \( Z'(4, 7) \)
Since none of the given response options match this exact calculation, let's verify the dilation for key examples and correct coordinates:
Let's check again:
1. For point X(0, 4):
- \( X' = (4 + 12(0 - 4), 7 + 12(4 - 7)) = (4 - 48, 7 - 36) = (-44, -29) \)
2. For point Y(2, 0):
- \( Y' = (4 + 12(2 - 4), 7 + 12(0 - 7)) = (4 - 24, 7 - 84) = (-20, -77) \)
3. For point Z(4, 7):
- \( Z' = (4 + 0, 7 + 0) = (4, 7) \)
The output of these calculations verifies the vertical changes in points. None of the original answers provided seem to reflect the outcome of the dilation correctly, so based on verified calculations, the coordinates for the vertices of triangle XYZ after dilation are \( X'(-44, -29) \), \( Y'(-20, -77) \), and \( Z'(4, 7) \).
If needing to select the closest match among options, ascertain they correspond theoretically to the progression if not by numbers.
1. **Identify the center of dilation**: The center is at point \( C(4, 7) \).
2. **Calculate the scale factor**: The scale factor is given as 12.
3. **Apply the dilation formula**: The formula to find the image point \( P' \) of point \( P(x, y) \) under dilation about a center \( C(a, b) \) with a scale factor \( k \) is given by:
\[
P' = \left( a + k(x - a), b + k(y - b) \right)
\]
4. **Apply the dilation to each vertex**:
- **Vertex \( X(0, 4) \)**:
\[
X' = \left( 4 + 12(0 - 4), 7 + 12(4 - 7) \right) = \left( 4 - 48, 7 - 36 \right) = (-44, -29)
\]
- **Vertex \( Y(2, 0) \)**:
\[
Y' = \left( 4 + 12(2 - 4), 7 + 12(0 - 7) \right) = \left( 4 - 24, 7 - 84 \right) = (-20, -77)
\]
- **Vertex \( Z(4, 7) \)**:
\[
Z' = \left( 4 + 12(4 - 4), 7 + 12(7 - 7) \right) = \left( 4 + 0, 7 + 0 \right) = (4, 7)
\]
5. **New Vertices**: The coordinates of the new vertices after dilation are:
- \( X'(-44, -29) \)
- \( Y'(-20, -77) \)
- \( Z'(4, 7) \)
Since none of the given response options match this exact calculation, let's verify the dilation for key examples and correct coordinates:
Let's check again:
1. For point X(0, 4):
- \( X' = (4 + 12(0 - 4), 7 + 12(4 - 7)) = (4 - 48, 7 - 36) = (-44, -29) \)
2. For point Y(2, 0):
- \( Y' = (4 + 12(2 - 4), 7 + 12(0 - 7)) = (4 - 24, 7 - 84) = (-20, -77) \)
3. For point Z(4, 7):
- \( Z' = (4 + 0, 7 + 0) = (4, 7) \)
The output of these calculations verifies the vertical changes in points. None of the original answers provided seem to reflect the outcome of the dilation correctly, so based on verified calculations, the coordinates for the vertices of triangle XYZ after dilation are \( X'(-44, -29) \), \( Y'(-20, -77) \), and \( Z'(4, 7) \).
If needing to select the closest match among options, ascertain they correspond theoretically to the progression if not by numbers.
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