Question

dilate quadrilateral abcd using center d and scale factor 2.

Explanation:

Step1: Recall Dilation Rules

Dilation about center \( D \) with scale factor \( 2 \) means for each point \( P \) (other than \( D \)) in the quadrilateral, the image \( P' \) satisfies \( \overrightarrow{DP'} = 2 \cdot \overrightarrow{DP} \). So we find vectors from \( D \) to \( A \), \( B \), \( C \), then double them.

Step2: Find Vector \( \overrightarrow{DA} \)

Let's assign coordinates (using the grid). Let \( D \) be at \( (x_D, y_D) \), \( A \) at \( (x_A, y_A) \). From the grid, if \( D \) is at, say, \( (d_x, d_y) \), \( A \) is 2 units right and 1 unit up from \( D \) (visually). So \( \overrightarrow{DA} = (2, 1) \). Doubling it: \( \overrightarrow{DP_A} = 2(2, 1) = (4, 2) \), so \( A' = D + (4, 2) \).

Step3: Find Vector \( \overrightarrow{DB} \)

From \( D \) to \( B \): visually, \( B \) is 3 units right and 1 unit down from \( D \)? Wait, better to count grid steps. Let's set \( D \) at \( (x, y) \). Let's take \( D \) as \( (2, 2) \) (for example, grid coordinates). Then \( A \) is \( (4, 3) \), \( B \) is \( (5, 1) \), \( C \) is \( (3, 0) \). Wait, maybe better to use vector approach. For point \( B \): vector from \( D \) to \( B \): \( (x_B - x_D, y_B - y_D) \). If \( D \) is \( (d_x, d_y) \), \( B \) is \( (d_x + 3, d_y - 1) \)? Wait, no, let's look at the grid. The distance from \( D \) to \( A \): horizontal steps: 2, vertical steps: 1. So scale factor 2: new horizontal from \( D \) to \( A' \) is \( 2 \times 2 = 4 \), vertical \( 2 \times 1 = 2 \). So \( A' \) is 4 right and 2 up from \( D \).

For \( C \): vector from \( D \) to \( C \): horizontal steps: 1, vertical steps: -2 (down 2). So doubling: horizontal 2, vertical -4. So \( C' \) is 2 right and 4 down from \( D \).

For \( B \): vector from \( D \) to \( B \): horizontal steps: 3, vertical steps: -1. Doubling: horizontal 6, vertical -2. So \( B' \) is 6 right and 2 down from \( D \).

Step4: Plot the Points

• \( D \) remains \( D \) (center of dilation).
• \( A' \): 4 right, 2 up from \( D \).
• \( B' \): 6 right, 2 down from \( D \).
• \( C' \): 2 right, 4 down from \( D \).

Then connect \( D \), \( A' \), \( B' \), \( C' \) to form the dilated quadrilateral.

Answer:

To dilate quadrilateral \( ABCD \) with center \( D \) and scale factor \( 2 \):

1. For each vertex \( A, B, C \) (excluding \( D \)):

• Draw a line from \( D \) through the vertex.
• Extend the line beyond the vertex so that the distance from \( D \) to the new vertex (\( A', B', C' \)) is twice the distance from \( D \) to the original vertex.

1. Connect \( D, A', B', C' \) to form the dilated quadrilateral.

(Visually, the new quadrilateral will have sides twice as long as the original, with \( D \) as the center, so \( DA' = 2 \cdot DA \), \( DB' = 2 \cdot DB \), \( DC' = 2 \cdot DC \), and the angles remain the same as the original quadrilateral.)