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? (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)
X was 2, 4
Y was 6, 0
Z was 7, 2
X was 2, 4
Y was 6, 0
Z was 7, 2
Answers
GPT-4o mini
To find the coordinates of the dilated triangle \(X'Y'Z'\) from triangle \(XYZ\) using a magnitude of \(2\) and a center of dilation at \((2, 0)\), we use the formula for dilation.
The formula for the coordinates after dilation is given by:
\[
X' = C + k(X - C)
\]
where:
- \(C\) is the center of dilation \((2, 0)\),
- \(k\) is the dilation factor (in this case \(k = 2\)),
- \((X, Y)\) are the original coordinates.
We apply this to each point \(X\), \(Y\), and \(Z\):
1. **For point \(X(2, 4)\)**:
- \(X' = (2, 0) + 2((2, 4) - (2, 0))\)
- \(= (2, 0) + 2(0, 4)\)
- \(= (2, 0) + (0, 8)\)
- \(= (2, 8)\)
2. **For point \(Y(6, 0)\)**:
- \(Y' = (2, 0) + 2((6, 0) - (2, 0))\)
- \(= (2, 0) + 2(4, 0)\)
- \(= (2, 0) + (8, 0)\)
- \(= (10, 0)\)
3. **For point \(Z(7, 2)\)**:
- \(Z' = (2, 0) + 2((7, 2) - (2, 0))\)
- \(= (2, 0) + 2(5, 2)\)
- \(= (2, 0) + (10, 4)\)
- \(= (12, 4)\)
Now that we have the coordinates for the dilated triangle \(X'Y'Z'\):
- \(X' = (2, 8)\)
- \(Y' = (10, 0)\)
- \(Z' = (12, 4)\)
The coordinates of the dilated triangle \(X'Y'Z'\) are \(X'(2, 8)\), \(Y'(10, 0)\), and \(Z'(12, 4)\).
The correct response is:
**X′(2,8), Y′(10,0), Z′(12,4)**.
The formula for the coordinates after dilation is given by:
\[
X' = C + k(X - C)
\]
where:
- \(C\) is the center of dilation \((2, 0)\),
- \(k\) is the dilation factor (in this case \(k = 2\)),
- \((X, Y)\) are the original coordinates.
We apply this to each point \(X\), \(Y\), and \(Z\):
1. **For point \(X(2, 4)\)**:
- \(X' = (2, 0) + 2((2, 4) - (2, 0))\)
- \(= (2, 0) + 2(0, 4)\)
- \(= (2, 0) + (0, 8)\)
- \(= (2, 8)\)
2. **For point \(Y(6, 0)\)**:
- \(Y' = (2, 0) + 2((6, 0) - (2, 0))\)
- \(= (2, 0) + 2(4, 0)\)
- \(= (2, 0) + (8, 0)\)
- \(= (10, 0)\)
3. **For point \(Z(7, 2)\)**:
- \(Z' = (2, 0) + 2((7, 2) - (2, 0))\)
- \(= (2, 0) + 2(5, 2)\)
- \(= (2, 0) + (10, 4)\)
- \(= (12, 4)\)
Now that we have the coordinates for the dilated triangle \(X'Y'Z'\):
- \(X' = (2, 8)\)
- \(Y' = (10, 0)\)
- \(Z' = (12, 4)\)
The coordinates of the dilated triangle \(X'Y'Z'\) are \(X'(2, 8)\), \(Y'(10, 0)\), and \(Z'(12, 4)\).
The correct response is:
**X′(2,8), Y′(10,0), Z′(12,4)**.