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 1/2 and a center of dilation at (4,7), what are the coordinate points of new vertices?

a. X' (-4,1), Y' (0,-7), Z' (4,7)
b. X' (2,5.5), Y' (3,3.5), Z' (4,7)
c. X' (-4,-3), Y' (-2,-7), Z' (0,0)
d. X' (4,11), Y' (6,7), Z' (8,14)

Answer 1

To find the new coordinates of the vertices after dilation, we will use the formula for dilation:

\[
X' = C + k(X - C)
\]

where:
- \(C\) is the center of dilation,
- \(k\) is the scale factor,
- \(X\) is the original point,
- \(X'\) is the dilated point.

In this case, the center of dilation \(C\) is at (4,7), and the scale factor \(k\) is \( \frac{1}{2} \).

Now let's apply this to each vertex of triangle XYZ.

### Vertex X (0,4):
1. Calculate the difference from the center:
\[
X - C = (0 - 4, 4 - 7) = (-4, -3)
\]
2. Apply the dilation formula:
\[
X' = (4, 7) + \frac{1}{2}(-4, -3) = (4, 7) + (-2, -1.5) = (4 - 2, 7 - 1.5) = (2, 5.5)
\]

### Vertex Y (2,0):
1. Calculate the difference from the center:
\[
Y - C = (2 - 4, 0 - 7) = (-2, -7)
\]
2. Apply the dilation formula:
\[
Y' = (4, 7) + \frac{1}{2}(-2, -7) = (4, 7) + (-1, -3.5) = (4 - 1, 7 - 3.5) = (3, 3.5)
\]

### Vertex Z (4,7):
1. Since point Z is the center of dilation, the coordinates remain unchanged:
\[
Z' = (4, 7)
\]

### Conclusion:
The new coordinates of triangle X'Y'Z' are:
- \(X' = (2, 5.5)\)
- \(Y' = (3, 3.5)\)
- \(Z' = (4, 7)\)

Reviewing the available choices, the closest set of dilated coordinates is option **b**: \(X' (2,5.5), Y' (3,3.5), Z' (4,7)\).