Question

identify the image for the following with the scale factor 2 and the center of dilation at the origin.

Explanation:

Step1: Find Coordinates of Original Triangle

First, identify the coordinates of the vertices of the original triangle \( \triangle XYZ \). From the graph:

• \( X \) is at \( (0, 0) \)
• \( Y \) is at \( (4, 0) \)
• \( Z \) is at \( (4, 2) \)

Step2: Apply Dilation with Scale Factor 2

The rule for dilation with scale factor \( k \) centered at the origin is \( (x, y) \to (kx, ky) \). For \( k = 2 \):

• For \( X(0, 0) \): \( (2 \times 0, 2 \times 0) = (0, 0) \)
• For \( Y(4, 0) \): \( (2 \times 4, 2 \times 0) = (8, 0) \)
• For \( Z(4, 2) \): \( (2 \times 4, 2 \times 2) = (8, 4) \)

Step3: Identify the Dilated Image

The dilated triangle will have vertices at \( (0, 0) \), \( (8, 0) \), and \( (8, 4) \). This triangle is similar to the original, with sides twice as long (since scale factor is 2) and centered at the origin.

Answer:

The image after dilation with scale factor 2 and center at the origin has vertices at \( (0, 0) \), \( (8, 0) \), and \( (8, 4) \) (a triangle with base length 8 and height 4, twice the original base of 4 and height of 2).