Question

given △abc, use a dilation with the center at the origin to make a similar triangle with side lengths three times as large. what are the coordinates of c of the image?
○ (-6, 3)
○ (-3, 6)
○ (3, -6)
○ (6, -3)

Explanation:

Step1: Find coordinates of C

From the graph, point C has coordinates \((2, -1)\)? Wait, no, looking at the grid: B is at (-1, -1)? Wait, no, let's re-examine. The grid: x-axis, y-axis. Let's see, point B: x=-1, y=-1? No, wait, the triangle: A is at (-1, 2), B is at (-1, -1), C is at (2, -1). Wait, no, maybe I misread. Wait, the grid lines: each square is 1 unit. Let's check the coordinates of C. From the graph, C is at (2, -1)? Wait, no, the original triangle: let's see, B is at (-1, -1), C is at (2, -1), A is at (-1, 2). Wait, no, maybe the coordinates of C are (2, -1)? Wait, no, the problem says dilation with center at origin, scale factor 3. Wait, maybe I made a mistake. Wait, looking at the options, the original C: let's check the graph again. Wait, the triangle: B is at (-1, -1)? No, maybe B is at (-1, 0)? Wait, no, the y-axis: the vertical line is y-axis, horizontal is x-axis. Let's see, point C: x=2, y=-1? Wait, no, the options have (6, -3), which is 3 times (2, -1). Wait, 23=6, -13=-3. So original C is (2, -1). Then dilation with scale factor 3, center at origin: (x, y) → (3x, 3y). So (23, -13) = (6, -3). Let's confirm.

Step2: Apply dilation formula

Dilation with center at origin and scale factor \( k \) transforms a point \( (x, y) \) to \( (kx, ky) \). Here, \( k = 3 \). First, find coordinates of \( C \). From the graph, \( C \) is at \( (2, -1) \). Then apply dilation: \( x' = 3 \times 2 = 6 \), \( y' = 3 \times (-1) = -3 \). So \( C' \) is \( (6, -3) \).

Answer:

D. (6, -3)