Question

use the drawing tool(s) to form the correct answer on the provided grid. triangle abc undergoes a sequence of transformations. 1. reflection across the line x = 1. 2. dilation by a factor of 2 centered at the origin. what is the resulting image?

Explanation:

Step1: Identify Original Coordinates

First, find the coordinates of triangle \( ABC \). From the graph:

• Let \( A = (2, 1) \), \( B = (2, -3) \), \( C = (7, -3) \) (assuming the grid and points' positions).

Step2: Reflect Across \( x = 1 \)

The formula for reflecting a point \( (x, y) \) across \( x = h \) is \( (2h - x, y) \). Here, \( h = 1 \).

• For \( A(2, 1) \): \( 2(1) - 2 = 0 \), so \( A' = (0, 1) \)
• For \( B(2, -3) \): \( 2(1) - 2 = 0 \), so \( B' = (0, -3) \)
• For \( C(7, -3) \): \( 2(1) - 7 = -5 \), so \( C' = (-5, -3) \)

Step3: Dilate by Factor 2 (Center Origin)

Dilation about origin: \( (x, y) \to (kx, ky) \), \( k = 2 \).

• For \( A'(0, 1) \): \( (2 \cdot 0, 2 \cdot 1) = (0, 2) \)
• For \( B'(0, -3) \): \( (2 \cdot 0, 2 \cdot (-3)) = (0, -6) \)
• For \( C'(-5, -3) \): \( (2 \cdot (-5), 2 \cdot (-3)) = (-10, -6) \)

Step4: Plot Dilated Points

Plot \( (0, 2) \), \( (0, -6) \), \( (-10, -6) \) and connect to form the dilated triangle.

Answer:

The resulting triangle after reflection and dilation has vertices at \( (0, 2) \), \( (0, -6) \), and \( (-10, -6) \) (drawn on the grid as per the transformation steps).