Question

1. dilate trapezoid wxyz

center of dilation: point z scale factor: 1/2
grid with trapezoid wxyz
w: ________ x: ________
y: ________ z: ________

Explanation:

Step1: Identify Coordinates of Original Points

First, we determine the coordinates of points \( W \), \( X \), \( Y \), and \( Z \) from the grid. Let's assume the grid has integer coordinates. From the diagram:

• \( Z \): Let's say \( Z = (1, 2) \) (assuming the origin is at the bottom-left, and each grid square is 1 unit). Wait, actually, looking at the grid, let's re - check. Let's set the coordinate system with the x - axis (horizontal) and y - axis (vertical). Let's assume the center of the grid lines:
• \( Z \): Let's find the coordinates. Let's say the horizontal (x) and vertical (y) positions. Let's assume \( Z=(1,2) \), \( W=(3,6) \), \( X=(5,6) \), \( Y=(9,2) \) (we need to confirm the grid. Wait, maybe a better way: the center of dilation is \( Z \), so for a dilation with center \( Z \) and scale factor \( \frac{1}{2} \), the formula for a point \( P(x,y) \) dilated about \( Z(a,b) \) with scale factor \( k \) is \( P'(a + k(x - a),b + k(y - b)) \).

Let's correctly identify the coordinates:
Looking at the grid, let's assume the coordinates:

• \( Z \): Let's say \( Z=(1,2) \) (x = 1, y = 2)
• \( W \): Let's see the horizontal distance from \( Z \): \( W \) is 2 units to the right (x - coordinate difference: \( 3 - 1=2 \)) and 4 units up (y - coordinate difference: \( 6 - 2 = 4 \)) from \( Z \). So \( W=(3,6) \)
• \( X \): \( X \) is 4 units to the right (x - coordinate difference: \( 5 - 1 = 4 \)) and 4 units up (y - coordinate difference: \( 6 - 2=4 \)) from \( Z \). So \( X=(5,6) \)
• \( Y \): \( Y \) is 8 units to the right (x - coordinate difference: \( 9 - 1=8 \)) and 0 units up (y - coordinate difference: \( 2 - 2 = 0 \)) from \( Z \). So \( Y=(9,2) \)

Step2: Apply Dilation Formula for Each Point

The formula for dilation of a point \( P(x,y) \) with center \( Z(a,b) \) and scale factor \( k \) is:
\( P'=(a + k(x - a),b + k(y - b)) \)
Since the center \( Z=(a,b)=(1,2) \) and \( k=\frac{1}{2} \)

For \( W(3,6) \):

\( x - a=3 - 1 = 2 \), \( y - b=6 - 2 = 4 \)
\( W'=(1+\frac{1}{2}(2),2+\frac{1}{2}(4))=(1 + 1,2 + 2)=(2,4) \)

For \( X(5,6) \):

\( x - a=5 - 1 = 4 \), \( y - b=6 - 2 = 4 \)
\( X'=(1+\frac{1}{2}(4),2+\frac{1}{2}(4))=(1 + 2,2 + 2)=(3,4) \)

For \( Y(9,2) \):

\( x - a=9 - 1 = 8 \), \( y - b=2 - 2 = 0 \)
\( Y'=(1+\frac{1}{2}(8),2+\frac{1}{2}(0))=(1 + 4,2+0)=(5,2) \)

For \( Z(1,2) \):

Since the center of dilation is \( Z \), when we dilate a point about itself, the image is the same as the original point. So \( Z'=(1,2) \)

Answer:

\( W':(2,4) \)
\( X':(3,4) \)
\( Y':(5,2) \)
\( Z':(1,2) \)