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′(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!**