Question

Use the image to answer the question.

A coordinate plane's x-axis ranges from negative 1 to 8 and its y-axis ranges from negative 2 to 5, both by 1-unit increments. 3 points forming a triangle are plotted, labeled upper X, upper Y, and upper Z, respectively.
X=(2, 4)
Y=(6, 0)
Z=(7, 2)
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′(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

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′(2,8), 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′(2,6), Y′(8,0), Z′(9.5,3)

Answer 1

To find the coordinates of the dilated triangle \(X'Y'Z'\) using a magnitude of 2 and a center of dilation at \((2,0)\), we can follow these steps:

1. **Determine the coordinates of the points before dilation:**
- \(X = (2, 4)\)
- \(Y = (6, 0)\)
- \(Z = (7, 2)\)

2. **Apply the dilation formula:**
For a point \(P(x, y)\), the coordinates after dilation \(P'(x', y')\) with center of dilation \(C(c_x, c_y)\) and a scale factor \(k\) can be found using the formula:

\[
x' = c_x + k \cdot (x - c_x)
\]
\[
y' = c_y + k \cdot (y - c_y)
\]

Here, \(C = (2, 0)\) and \(k = 2\).

3. **Calculate the new coordinates for each point:**

- **For point \(X(2, 4)\):**
\[
x' = 2 + 2 \cdot (2 - 2) = 2 + 0 = 2
\]
\[
y' = 0 + 2 \cdot (4 - 0) = 0 + 8 = 8
\]
Thus, \(X' = (2, 8)\).

- **For point \(Y(6, 0)\):**
\[
x' = 2 + 2 \cdot (6 - 2) = 2 + 2 \cdot 4 = 2 + 8 = 10
\]
\[
y' = 0 + 2 \cdot (0 - 0) = 0 + 0 = 0
\]
Thus, \(Y' = (10, 0)\).

- **For point \(Z(7, 2)\):**
\[
x' = 2 + 2 \cdot (7 - 2) = 2 + 2 \cdot 5 = 2 + 10 = 12
\]
\[
y' = 0 + 2 \cdot (2 - 0) = 0 + 4 = 4
\]
Thus, \(Z' = (12, 4)\).

4. **Final coordinates of the dilated triangle \(X'Y'Z'\):**
- \(X' = (2, 8)\)
- \(Y' = (10, 0)\)
- \(Z' = (12, 4)\)

5. **Comparing to the responses provided:**
The matching response for the dilated triangle's coordinates is:
\[
X'(2,8), Y'(10,0), Z'(12,4)
\]

**Final Answer:**
- \(X'(2, 8), Y'(10, 0), Z'(12, 4)\)