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′(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′(4,11)
, 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,−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 determine the new coordinates of the vertices of triangle XYZ after dilation, we need to apply the dilation transformation using the center of dilation at point (4,7) with a magnitude of 12.

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

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

Here, we have:

• Center of dilation \((x_c, y_c) = (4, 7)\)
• Scale factor \(k = 12\)

Now, we will apply this formula to each vertex of triangle XYZ.

1. Vertex X(0, 4):

\[ x' = 4 + 12 \cdot (0 - 4) = 4 + 12 \cdot -4 = 4 - 48 = -44 \] \[ y' = 7 + 12 \cdot (4 - 7) = 7 + 12 \cdot -3 = 7 - 36 = -29 \] So, \(X'(-44, -29)\).

2. Vertex Y(2, 0):

\[ x' = 4 + 12 \cdot (2 - 4) = 4 + 12 \cdot -2 = 4 - 24 = -20 \] \[ y' = 7 + 12 \cdot (0 - 7) = 7 + 12 \cdot -7 = 7 - 84 = -77 \] So, \(Y'(-20, -77)\).

3. Vertex Z(4, 7):

\[ x' = 4 + 12 \cdot (4 - 4) = 4 + 12 \cdot 0 = 4 \] \[ y' = 7 + 12 \cdot (7 - 7) = 7 + 12 \cdot 0 = 7 \] So, \(Z'(4, 7)\).

Now, we have the coordinates of the new vertices:

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

However, this does not appear to match any of the provided options. It seems there may have been a misunderstanding in the application of the dilation magnitude. Since the calculations seem to take the dilation too far from the center point, we might want to double-check for any errors in the calculations.

Let's check one more example using a smaller example to verify what we would expect:

Revisiting the inputs: The original coordinates:

• X(0, 4)
• Y(2, 0)
• Z(4, 7)

Center of Dilation = (4,7) Scale factor = 12.

Calculating:

1. Recalculating X(0, 4): \[ x' = 4 + 12 \cdot (0 - 4) = 4 - 48 = -44 \] \[ y' = 7 + 12 \cdot (4 - 7) = 7 - 36 = -29 \]

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

3. The point Z is unchanged, remaining at: \(Z' = (4, 7)\).

Summary of new points:

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

Since the computed coordinate points do not match provided options, it suggests there may need to be clarification on any configuration in the workspace or the visibility of options available.

Please confirm the conditions set if misaligned or need more investigation on points chosen!