Question

use the drop - down menus to describe the transformations used to map $\triangle abc$ onto $\triangle abc$

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

the triangles are
(there is a coordinate plane image with $\triangle abc$ and $\triangle abc$ plotted, and a calculator icon)

Explanation:

Step1: Analyze Reflection Axis

To determine the reflection axis, observe the coordinates of the original triangle \( \triangle ABC \) and the transformed \( \triangle A'B'C' \). The original triangle is above the \( x \)-axis, and the transformed one is below, indicating a reflection across the \( x \)-axis? Wait, no—wait, looking at the \( y \)-coordinates: original points have positive \( y \), transformed have negative \( y \)? Wait, no, let's check the graph. Wait, the original \( \triangle ABC \): point \( B \) is at \( (-5, 4) \), \( A \) at \( (3, 5) \), \( C \) at \( (4, 3) \)? Wait, no, the grid: \( B \) is at \( (-5, 4) \), \( A \) at \( (3, 5) \), \( C \) at \( (4, 3) \). After dilation (scale factor 2), the dilated points would be \( B'(-10, 8) \), \( A'(6, 10) \), \( C'(8, 6) \)? But the transformed triangle \( \triangle A'B'C' \) has \( B' \) at \( (-10, -8) \), \( A'(6, -10) \), \( C'(8, -6) \)? Wait, no, the graph shows \( B' \) at \( (-10, -8) \), \( A' \) at \( (6, -10) \), \( C' \) at \( (8, -6) \)? Wait, no, the lower triangle: \( B' \) is at \( (-10, -8) \), \( A' \) at \( (6, -10) \), \( C' \) at \( (8, -6) \)? Wait, no, the original \( \triangle ABC \) is in the upper left/right, and \( \triangle A'B'C' \) is in the lower left/right. Wait, the reflection across the \( x \)-axis would flip \( y \)-coordinates. Let's check a point: original \( B(-5, 4) \), after dilation (scale 2) is \( (-10, 8) \), then reflection across \( x \)-axis would be \( (-10, -8) \), which matches \( B' \) in the graph. So the reflection is across the \( x \)-axis? Wait, but the dropdown has \( x \)-axis and \( y \)-axis. Wait, the original triangle and the transformed: the \( x \)-coordinates of corresponding points (after dilation) have the same sign? Wait, \( B \) is at \( (-5, 4) \), dilated \( (-10, 8) \), reflected across \( x \)-axis: \( (-10, -8) \), which is \( B' \). So the reflection is across the \( x \)-axis? Wait, but the dropdown menu in the problem shows "x-axis" and "y-axis". Wait, the problem's dropdown for the second transformation: the options are \( x \)-axis or \( y \)-axis. Wait, the original triangle and the transformed: the \( y \)-coordinates are flipped in sign (positive to negative), so reflection across \( x \)-axis. Wait, but let's confirm: reflection across \( x \)-axis changes \( (x, y) \) to \( (x, -y) \). After dilation (scale 2), \( (x, y) \to (2x, 2y) \), then reflection across \( x \)-axis: \( (2x, -2y) \). So the second transformation is a reflection across the \( x \)-axis? Wait, but the dropdown in the problem has "x-axis" and "y-axis" as options. Wait, the user's problem: the second dropdown is for reflection across which axis. So the correct axis is \( x \)-axis? Wait, no—wait, looking at the graph, the original triangle is above the \( x \)-axis, and the transformed is below, so reflection across \( x \)-axis. Wait, but the dropdown in the problem shows "x-axis" and "y-axis". So the answer for the second transformation is reflection across the \( x \)-axis? Wait, but let's check again. Wait, the original \( \triangle ABC \): points have positive \( y \)-coordinates, transformed \( \triangle A'B'C' \) have negative \( y \)-coordinates, so reflection across \( x \)-axis. So the second transformation is a reflection across the \( x \)-axis? Wait, but the dropdown in the problem: the options are \( x \)-axis and \( y \)-axis. So the correct choice is \( x \)-axis? Wait, no—wait, maybe I made a mistake. Wait, the dilation is centered at the origin with scale factor 2, then reflection. Let's take point \( B \): original \( (-5, 4) \), after dilation: \( (-10, 8) \), after reflection across \( x \)-axis: \( (-10, -8) \), which matches the \( B' \) in the graph. So the reflection is across the \( x \)-axis.

Step2: Confirm the Reflection Axis

By comparing the \( y \)-coordinates of corresponding points (after dilation) and the transformed points, we see that the \( y \)-coordinate sign flips (positive to negative), which is characteristic of a reflection across the \( x \)-axis. Thus, the second transformation is a reflection across the \( x \)-axis.

Answer:

For the second transformation, the reflection is across the \( \boldsymbol{x} \)-axis. So the dropdown should be selected as "x-axis".