Question

using dilations to make similar figures
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 triangle: B is at (-1, -1)? Wait, no, the grid: x-axis, y-axis. Let's see, point C: looking at the graph, the original triangle: B is at (-1, -1)? Wait, no, the green triangle: A is at (-1, 2), B is at (-1, -1), and C is at (2, -1). Wait, yes, because from B to C, it's along y=-1, x from -1 to 2, so C is (2, -1). Wait, no, the grid lines: each square is 1 unit. So B is at (-1, -1)? Wait, no, the y-axis is at x=0. So B is at (-1, -1)? Wait, A is at (-1, 2), B is at (-1, -1), and C is at (2, -1). Yes, because the base is from x=-1 to x=2 (length 3) and height from y=-1 to y=2 (length 3). So C is (2, -1).

Step2: Apply dilation with scale factor 3

Dilation with center at origin (0,0) and scale factor \(k = 3\) means we multiply each coordinate by 3. So for a point \((x, y)\), the image \((x', y') = (k \cdot x, k \cdot y)\). So for C(2, -1), the image \(C'\) will be \((3 \cdot 2, 3 \cdot (-1)) = (6, -3)\)? Wait, no, wait, maybe I got C's coordinates wrong. Wait, let's check again. Wait, the original triangle: looking at the graph, B is at (-1, -1)? No, maybe B is at (-1, -1)? Wait, no, the y-axis is 0. Let's see the x-coordinate of C: from the graph, C is at (2, -1)? Wait, no, the options include (6, -3), (3, -6), etc. Wait, maybe I messed up C's coordinates. Wait, let's look at the options. The options are (-6,3), (-3,6), (3,-6), (6,-3). So let's re-express. Maybe the original C is (2, -1)? No, wait, maybe the original C is (2, -1)? Wait, no, maybe the original C is (2, -1), but when we dilate by 3, (23, -13) = (6, -3), which is one of the options (6, -3). Wait, let's confirm:

Wait, maybe the original C is (2, -1)? Wait, no, let's check the graph again. The triangle: A is at (-1, 2), B is at (-1, -1), C is at (2, -1). So C is (2, -1). Dilation with scale factor 3: multiply x and y by 3. So \(x' = 2 \times 3 = 6\), \(y' = -1 \times 3 = -3\). So C' is (6, -3), which is option D: (6, -3).

Wait, but let's check again. Maybe I had C's coordinates wrong. Let's see: if the original C is (2, -1), then dilation by 3 gives (6, -3). Which is option (6, -3), one of the choices. So that's the answer.

Answer:

(6, -3) (corresponding to the option D: (6, -3))