Triangle XYZ

Question

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,−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)
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′(2,5.5)
, Y′(3,3.5)
, Z′(4,7)

Answer 1

To find the coordinates of the new vertices of triangle \(X'Y'Z'\) after dilating triangle \(XYZ\) with a center of dilation at \((4,7)\) and a magnitude of 12, we can use the formula for dilation:

\[ X' = (kx + (1-k)h, ky + (1-k)k) \]

where:

• \((x, y)\) are the original coordinates of the vertex,
• \((h, k)\) is the center of dilation,
• \(k\) is the scale factor.

Since we are using a magnitude (scale factor) of 12, \(k = 12\).

Let's calculate the new coordinates of each vertex using the center of dilation at \((4, 7)\):

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

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

3. For vertex \(Z(4, 7)\): \[ Z' = (12 * (4 - 4) + 4, 12 * (7 - 7) + 7) \] \[ = (12*(0) + 4, 12*(0) + 7) = (4, 7) \]

After calculating, the new coordinates of the vertices are:

• \(X'(-44, -29)\)
• \(Y'(-20, -77)\)
• \(Z'(4, 7)\) (remains the same since it's the center of dilation)

From the answer choices, none of these coordinates match the responses given. Could you please double-check the scale factor or the center of dilation?