Question

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′(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′(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,−3), Y′(−2,−7), Z′(0,0) upper X prime left parenthesis negative 4 comma negative 3 right parenthesis , upper Y prime left parenthesis negative 2 comma negative 7 right parenthesis , upper Z prime left parenthesis 0 comma 0 right parenthesis X′(−4,1), Y′(0,−7), Z′(4,7)

Answer 1

To find the new vertices after dilation of triangle XYZ with a center at (4,7) and a magnitude of 12, we can use the dilation formula:

\[
X' = C + k(X - C)
\]

Where:
- \(X'\) is the new point after dilation,
- \(C\) is the center of dilation,
- \(k\) is the scale factor,
- \(X\) is the original point.

Using the given center of dilation \(C(4,7)\) and scale factor \(k = 12\), we will calculate the new vertex positions for points \(X(0,4)\), \(Y(2,0)\), and \(Z(4,7)\).

### Calculating new points:

1. For vertex \(X(0,4)\):
\[
X' = (4,7) + 12 \cdot ((0,4) - (4,7))
\]
\[
= (4,7) + 12 \cdot (-4,-3)
\]
\[
= (4,7) + (-48, -36)
\]
\[
= (4 - 48, 7 - 36) = (-44, -29)
\]

2. For vertex \(Y(2,0)\):
\[
Y' = (4,7) + 12 \cdot ((2,0) - (4,7))
\]
\[
= (4,7) + 12 \cdot (-2,-7)
\]
\[
= (4,7) + (-24, -84)
\]
\[
= (4 - 24, 7 - 84) = (-20, -77)
\]

3. For vertex \(Z(4,7)\) (since the center of dilation is at \(Z\)):
\[
Z' = (4,7) + 12 \cdot ((4,7) - (4,7))
\]
\[
= (4,7) + 12 \cdot (0,0) = (4,7)
\]

Now, we have the coordinates of the new vertices:
- \(X'(-44,-29)\)
- \(Y'(-20,-77)\)
- \(Z'(4,7)\)

Given the options for responses, none of them correspond to our calculated results. Please check if the dilation center or magnitude was misinterpreted or if additional context is provided. The dilation should preserve the linear relationships; we expect the calculated points to be fairly reachable when recalibrated based on understood terms.