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)
graph of triangle abc on a coordinate plane with x - axis from - 4 to 4, y - axis from - 4 to 4, grid lines, points a, b, c labeled

Explanation:

Step1: Find coordinates of C

From the graph, point C has coordinates \((2, -1)\) (assuming the grid and position, let's confirm: looking at the x - axis, C is at x = 2, y - axis, since it's on the line just below the x - axis, y=-1? Wait, no, looking at the triangle, B is at (-1, -1)? Wait, no, let's re - examine. Wait, the original triangle: B is at (-1, -1)? No, looking at the grid, the x - axis and y - axis. Let's see, the original point C: from the graph, the coordinates of C are \((2, -1)\)? Wait, no, maybe I made a mistake. Wait, the triangle is drawn with B at (-1, -1)? No, looking at the grid lines, each square is 1 unit. Let's see, point B: x=-1, y = - 1? Point A: x=-1, y = 2? Point C: x = 2, y=-1? Wait, no, maybe the coordinates of C are \((2, -1)\)? Wait, no, let's check again. Wait, the problem says "dilation with center at the origin to make a similar triangle with side lengths three times as large". The rule for dilation with center at the origin is \((x,y)\to(kx,ky)\) where \(k\) is the scale factor. Here, \(k = 3\). First, find the coordinates of C. From the graph, looking at the grid, point C is at \((2,-1)\)? Wait, no, maybe I misread. Wait, the triangle: B is at (-1, -1), A is at (-1, 2), and C is at (2, -1). So the coordinates of C are \((2,-1)\).

Step2: Apply dilation with k = 3

The formula for dilation with center at the origin is \((x,y)\to(3x,3y)\). For point C with coordinates \((2,-1)\), when we apply the dilation, the new x - coordinate is \(3\times2=6\) and the new y - coordinate is \(3\times(- 1)=-3\). So the coordinates of \(C'\) are \((6,-3)\).

Answer:

\((6, - 3)\) (corresponding to the option D. \((6, - 3)\) if we consider the options as A. \((-6,3)\), B. \((-3,6)\), C. \((3,-6)\), D. \((6,-3)\))