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′(4,11)
, 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′(2,5.5)
, Y′(3,3.5)
, Z′(4,7)

Answer 1

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

The formula for dilation about a point \((x_c, y_c)\) with a scale factor \(k\) is given by:

\[ X' = (x_c + k \cdot (x - x_c), y_c + k \cdot (y - y_c)) \]

where \((x, y)\) are the original coordinates and \((x', y')\) are the new coordinates after dilation.

Let's apply this to each vertex:

1. For point X(0, 4):

   • Substitute into the formula: \[ X' = \left(4 + 12 \cdot (0 - 4), 7 + 12 \cdot (4 - 7)\right) \] \[ X' = \left(4 - 48, 7 - 36\right) = (-44, -29) \]

2. For point Y(2, 0):

   • Substitute into the formula: \[ Y' = \left(4 + 12 \cdot (2 - 4), 7 + 12 \cdot (0 - 7)\right) \] \[ Y' = \left(4 - 24, 7 - 84\right) = (-20, -77) \]

3. For point Z(4, 7):

   • Substitute into the formula: \[ Z' = \left(4 + 12 \cdot (4 - 4), 7 + 12 \cdot (7 - 7)\right) \] \[ Z' = (4, 7) \]

After calculating the above points, we observe that a dilatation with the center of dilation matching vertex Z results in the coordinates of Z' remaining unchanged at (4, 7).

As we check through the provided options, it seems that a mistake was made while calculating, so let's simplify the calculation for each vertex based on the above ones:

1. For X(0, 4): \[ X' = \left(4 + 12 \left(0 - 4\right), 7 + 12 \left(4 - 7\right)\right) = (-44, -29) \]

2. For Y(2, 0): \[ Y' = \left(4 + 12 \left(2 - 4\right), 7 + 12 \left(0 - 7\right)\right) = (-20, -77) \]

3. Z remains (4, 7).

Combining all this, it seems there was a confusion with the dilations' center and the way coordinates transformed.

After confirming we derived the displacement magnitudes correctly, we return to the supplied options. No direct answer fits derived points well, so please provide correct dilation parameters or review given options as they appear misaligned with a standard dilation specification given.

For the coordinates from the provided options:

• As calculated those coordinates above with coordinates do not fit, Hence Please verify as points calculated via systematic dilation mechanics confirm a diverging endpoint.