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′(4,11)
, 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′(−4,1)
, 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,−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′(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 dilation with a center of dilation at (4,7) and a magnitude of 12, we can follow these steps for each vertex:

1. Subtract the center of dilation coordinates from the vertex coordinates.
2. Multiply the results by the scale factor (magnitude of dilation).
3. Add the center of dilation coordinates back to the new coordinate.

Given:

• Dilation center: C(4,7)
• Magnitude of dilation: 12

Calculating for each vertex:

For vertex X(0,4):

1. Subtract the center: (0 - 4, 4 - 7) = (-4, -3)
2. Multiply by 12: (-4 * 12, -3 * 12) = (-48, -36)
3. Add the center back: (-48 + 4, -36 + 7) = (-44, -29)

For vertex Y(2,0):

1. Subtract the center: (2 - 4, 0 - 7) = (-2, -7)
2. Multiply by 12: (-2 * 12, -7 * 12) = (-24, -84)
3. Add the center back: (-24 + 4, -84 + 7) = (-20, -77)

For vertex Z(4,7):

1. Subtract the center: (4 - 4, 7 - 7) = (0, 0)
2. Multiply by 12: (0 * 12, 0 * 12) = (0, 0)
3. Add the center back: (0 + 4, 0 + 7) = (4, 7)

So the new coordinates of the vertices after dilation are:

• X'(-44, -29)
• Y'(-20, -77)
• Z'(4, 7)

However, it looks like my calculations result in coordinates that do not match any of the responses provided. Let's clarify the dilation correctly.

Correction Process: Let's use the dilation formula directly:

1. New point = C + k * (P - C), where C is the center, k is the scale factor, and P is the original point.

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

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

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

So the final coordinates after correction yield:

• X'(-44, -29)
• Y'(-20, -77)
• Z'(4, 7)

The new coordinates still do not match the responses and must have been misunderstood in notation. Make sure to check the provided options again or rethink the problem setup on vertex locations properly considering the contributions.

But based on basic dilation principles used, none of the presented options correctly reflect results falling from accurate dilation.