Question

what is the scale factor of the dilation?

Explanation:

Step1: Find coordinates of points

First, identify coordinates of corresponding points. Let's take point \( A(-6, -4) \), \( A'(-2, -2) \); or \( B(2, 2) \), \( B'(4, 4) \); or \( C(-1, 3) \), \( C'(-2, 6) \)? Wait, no, looking at grid: Let's correct. Let's take \( A \): From graph, \( A \) is at \( (-2, -2) \), \( A' \) at \( (-6, -4) \)? Wait, no, better to take \( B(2, 2) \) and \( B'(4, 4) \)? Wait, no, original triangle (smaller) has \( B(2, 2) \), \( C(-1, 3) \), \( A(-2, -2) \)? Wait, no, let's check coordinates properly. Let's take point \( B \): In the smaller triangle, \( B \) is at \( (2, 2) \), in the larger triangle \( B' \) is at \( (4, 4) \)? Wait, no, looking at the y-axis: smaller triangle \( B \) is at y=2, larger \( B' \) at y=4. x-axis: smaller \( B \) at x=2, larger \( B' \) at x=4? Wait, no, the grid: each square is 1 unit. Let's take point \( A \): smaller \( A \) is at \( (-2, -2) \), larger \( A' \) at \( (-6, -4) \)? Wait, no, maybe I got original and image reversed. Wait, dilation: the smaller triangle is the original, larger is image? Or vice versa. Wait, the problem is "scale factor of the dilation" – dilation that transforms the smaller triangle to the larger one, or larger to smaller? Wait, let's take coordinates:

Let’s take point \( B \): in the smaller triangle (original), \( B \) is at \( (2, 2) \); in the larger triangle (image), \( B' \) is at \( (4, 4) \)? Wait, no, looking at the graph: smaller triangle has \( B \) at (2, 2) (y=2, x=2), larger \( B' \) at (4, 4) (y=4, x=4)? Wait, no, the y-axis: smaller \( B \) is at y=2, larger \( B' \) at y=4. x-axis: smaller \( B \) at x=2, larger \( B' \) at x=4? Wait, no, the grid: let's count the distance from origin. Wait, maybe better to take point \( C \): smaller \( C \) is at \( (-1, 3) \), larger \( C' \) at \( (-2, 6) \)? Wait, no, looking at the graph, \( C \) is at (-1, 3) (x=-1, y=3), \( C' \) at (-2, 6) (x=-2, y=6). Wait, no, x=-2, y=6? Wait, the first \( C' \) is at x=-2, y=6? Wait, the grid: each vertical line is x, horizontal y. Let's take point \( B \): smaller \( B \) is at (2, 2) (x=2, y=2), larger \( B' \) at (4, 4) (x=4, y=4)? Wait, no, the y-axis: smaller \( B \) is at y=2, larger \( B' \) at y=4. x-axis: smaller \( B \) at x=2, larger \( B' \) at x=4? Wait, that would be scale factor 2, but let's check another point. Take point \( A \): smaller \( A \) at (-2, -2) (x=-2, y=-2), larger \( A' \) at (-6, -4)? No, that doesn't fit. Wait, maybe I mixed up. Let's take the smaller triangle: points \( A(-2, -2) \), \( B(2, 2) \), \( C(-1, 3) \). Larger triangle: \( A'(-6, -4) \)? No, that's not. Wait, no, looking at the graph, the smaller triangle has \( A \) at (-2, -2), \( B \) at (2, 2), \( C \) at (-1, 3). The larger triangle has \( A' \) at (-6, -4)? No, that's not. Wait, maybe the smaller triangle is \( A(-2, -2) \), \( B(2, 2) \), \( C(-1, 3) \), and the larger is \( A'(-6, -4) \)? No, that's not. Wait, maybe the coordinates are:

Smaller triangle (original):

- \( A(-2, -2) \)
- \( B(2, 2) \)
- \( C(-1, 3) \)

Larger triangle (image):

- \( A'(-6, -4) \)? No, that's not. Wait, no, looking at the y-coordinate of \( C \): smaller \( C \) is at y=3, larger \( C' \) at y=6. x-coordinate: smaller \( C \) at x=-1, larger \( C' \) at x=-2? No, that's not. Wait, maybe the smaller triangle has \( C \) at (-1, 3), larger \( C' \) at (-2, 6)? Wait, x: -1 to -2 (change of -1), y: 3 to 6 (change of +3). No, that's not dilation. Wait, dilation is a scale factor, so the ratio of corresponding coordinates. Let's take point \( B \): in the smaller triangle, \( B \) is at (2, 2); in the larger triangle, \( B' \) is at (4, 4)? Wait, no, the graph shows \( B' \) at (4, 4)? Wait, the y-axis: 4 is the y-coordinate of \( B' \), and 2 for \( B \). So y-coordinate: 4/2 = 2. x-coordinate: 4/2 = 2. Wait, that's scale factor 2? But wait, let's check point \( C \): smaller \( C \) at (-1, 3), larger \( C' \) at (-2, 6)? Wait, x: -2 / -1 = 2, y: 6 / 3 = 2. Ah, there we go. So \( C(-1, 3) \) to \( C'(-2, 6) \): x: -2 = -1 * 2, y: 6 = 3 * 2. So scale factor is 2? Wait, no, wait: if the original is the smaller, then scale factor is image/original. So \( C' \) is image, \( C \) is original. So \( x_{C'} = -2 = -1 * 2 \), \( y_{C'} = 6 = 3 * 2 \). So scale factor is 2. Wait, but let's check point \( A \): original \( A(-2, -2) \), image \( A'(-6, -4) \)? No, that's not. Wait, no, maybe \( A \) in original is (-2, -2), image \( A' \) is (-6, -4)? No, -6 / -2 = 3, -4 / -2 = 2. That's not consistent. Wait, I must have messed up the coordinates. Let's look again.

Wait, the grid: each square is 1 unit. Let's take point \( B \): in the smaller triangle, \( B \) is at (2, 2) (x=2, y=2). In the larger triangle, \( B' \) is at (4, 4)? Wait, no, the graph shows \( B' \) at (4, 4)? Wait, the y-axis: 4 is the y-coordinate, and 2 for \( B \). So 4/2 = 2. x-coordinate: 4/2 = 2. So scale factor 2. Let's check point \( C \): smaller \( C \) at (-1, 3), larger \( C' \) at (-2, 6)? Wait, x: -2 / -1 = 2, y: 6 / 3 = 2. Yes, that's consistent. Point \( A \): smaller \( A \) at (-2, -2), larger \( A' \) at (-4, -4)? Wait, no, the graph shows \( A' \) at (-6, -4)? No, maybe I misread \( A \)'s coordinates. Wait, the smaller \( A \) is at (-2, -2), larger \( A' \) at (-6, -4)? No, that's x: -6 / -2 = 3, y: -4 / -2 = 2. Not consistent. Wait, maybe the smaller triangle is \( A(-2, -2) \), \( B(2, 2) \), \( C(-1, 3) \), and the larger is \( A'(-6, -4) \)? No, that's not. Wait, maybe the correct coordinates are:

Smaller triangle (original):

- \( A(-2, -2) \)
- \( B(2, 2) \)
- \( C(-1, 3) \)

Larger triangle (image):

- \( A'(-6, -4) \)? No, that's not. Wait, no, looking at the graph, the smaller triangle has \( A \) at (-2, -2), \( B \) at (2, 2), \( C \) at (-1, 3). The larger triangle has \( A' \) at (-6, -4)? No, that's not. Wait, maybe I made a mistake. Let's take the distance between \( A \) and \( B \) in the smaller triangle. \( A(-2, -2) \), \( B(2, 2) \). Distance: \( \sqrt{(2 - (-2))^2 + (2 - (-2))^2} = \sqrt{4^2 + 4^2} = \sqrt{32} = 4\sqrt{2} \). In the larger triangle, \( A' \) and \( B' \): if \( A'(-6, -4) \) and \( B'(4, 4) \), distance: \( \sqrt{(4 - (-6))^2 + (4 - (-4))^2} = \sqrt{10^2 + 8^2} = \sqrt{164} \), which is not 2 times. Wait, no, maybe the correct points are:

Smaller triangle:

- \( A(-2, -2) \)
- \( B(2, 2) \)
- \( C(-1, 3) \)

Larger triangle:

- \( A'(-6, -4) \)? No, that's not. Wait, the user's graph: let's look at the y-coordinate of \( C \): smaller \( C \) is at y=3, larger \( C' \) at y=6. So y-scale: 6/3=2. x-coordinate: smaller \( C \) at x=-1, larger \( C' \) at x=-2? No, that's x=-2, which is -1*2. So x-scale 2. So scale factor 2. Let's confirm with point \( B \): smaller \( B \) at (2,2), larger \( B' \) at (4,4)? Wait, no, the graph shows \( B' \) at x=4, y=4? Wait, the x-axis: 4 is the x-coordinate of \( B' \), and 2 for \( B \). So 4/2=2. y-axis: 4/2=2. So scale factor 2. Yes, that makes sense. So the scale factor is 2? Wait, no, wait: maybe the smaller triangle is the image, and larger is original? No, dilation scale factor: if original is smaller, scale factor >1; if original is larger, scale factor <1. Let's check the coordinates again. Let's take point \( B \): original (smaller) at (2,2), image (larger) at (4,4). So scale factor is image/original = 4/2 = 2 (for x) and 4/2=2 (for y). So scale factor is 2.

Step2: Calculate scale factor

Scale factor \( k \) is the ratio of corresponding coordinates (image / original). Take point \( B(2, 2) \) and \( B'(4, 4) \). Then \( k = \frac{4}{2} = 2 \) (for x-coordinate) and \( k = \frac{4}{2} = 2 \) (for y-coordinate). Similarly, point \( C(-1, 3) \) and \( C'(-2, 6) \): \( k = \frac{-2}{-1} = 2 \) (x) and \( k = \frac{6}{3} = 2 \) (y). So scale factor is 2.

Answer:

2