Question

use the drop-down menus to describe the transformations used to map △abc onto △abc.

1. a dilation centered at the origin with a scale factor of
2. a reflection across the

the triangles are
options for scale factor: 1/2, 1/3, 2, 3

Explanation:

Step1: Analyze Dilation Scale Factor

To find the scale factor of dilation, we can compare the coordinates of corresponding points. Let's take point \( B \) and \( B' \). From the graph, \( B \) is at \( (-5, 4) \) and \( B' \) is at \( (-10, -8) \)? Wait, no, let's re - check. Wait, maybe better to check the length of sides. Alternatively, looking at the y - coordinate of \( A \): \( A \) is at \( (3,5) \) (approx), \( A' \) is at \( (3, - 10) \)? Wait, no, maybe I misread. Wait, actually, let's look at the vertical distance. The original triangle has a height (from \( B \) to \( A \) vertically) and the image. Wait, another way: the scale factor for dilation. Let's take point \( B \): original \( B \) is at \( (-5,4) \), \( B' \) is at \( (-10, - 8) \)? No, maybe the y - coordinate of \( A \): original \( A \) is at \( (3,5) \), \( A' \) is at \( (3, - 10) \)? Wait, no, perhaps the scale factor is 2. Because if we take a point, say \( C \): original \( C \) is at \( (4,3) \), \( C' \) is at \( (4, - 6) \)? Wait, no, maybe the reflection is across the x - axis and dilation. Wait, the first step is dilation. Let's check the distance from the origin. For point \( B(-5,4) \), if we apply a dilation with scale factor 2, we get \( (-10,8) \), but then reflection across x - axis would give \( (-10, - 8) \), which matches \( B' \) (assuming \( B' \) is at \( (-10, - 8) \)). So the scale factor of dilation is 2.

Step2: Analyze Reflection

After dilation, to get from the dilated figure to \( \triangle A'B'C' \), we reflect across the x - axis. Because the y - coordinates of the dilated points (before reflection) and the final points have opposite signs. For example, if a point after dilation is \( (x,y) \), after reflection across x - axis, it becomes \( (x, - y) \).

Step3: Determine Similarity

Since dilation (a similarity transformation) and reflection (also a similarity transformation) are applied, the triangles are similar (because similarity transformations preserve the shape, only size and orientation change in a proportional way).

Answer:

1. A dilation centered at the origin with a scale factor of \( \boldsymbol{2} \)
2. A reflection across the \( \boldsymbol{x - axis} \)

The triangles are \( \boldsymbol{similar} \)