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. a reflection across the

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

Explanation:

Step1: Analyze Dilation Scale Factor

First, we find the coordinates of the original triangle \( \triangle ABC \) and the transformed triangle \( \triangle A'B'C' \). Let's assume the coordinates:

• For \( \triangle ABC \): Let's say \( B(-5, 2) \), \( C(4, 2) \), \( A(2, 4) \) (from the graph).
• For \( \triangle A'B'C' \): \( B'(-10, -4) \), \( C'(8, -4) \), \( A'(4, -8) \).

To find the scale factor of dilation, we take the ratio of the distance from the origin to a corresponding point in \( \triangle A'B'C' \) and \( \triangle ABC \). For point \( B \): original \( x = -5 \), transformed \( x = -10 \). The ratio \( \frac{-10}{-5} = 2 \). Similarly for \( y \)-coordinate: original \( y = 2 \), transformed \( y = -4 \), but considering dilation (scale factor affects both axes), the scale factor is \( 2 \) (since \( -10 = -5 \times 2 \), \( -4 = 2 \times (-2) \), but dilation scale factor is positive, and reflection will handle the sign change). So the scale factor for dilation is \( 2 \).

Step2: Analyze Reflection Axis

Looking at the \( y \)-coordinates: original \( \triangle ABC \) has positive \( y \)-values, transformed \( \triangle A'B'C' \) has negative \( y \)-values. So the reflection is across the \( x \)-axis (since reflecting across \( x \)-axis changes \( (x, y) \) to \( (x, -y) \)).

Step3: Determine Similarity

Since dilation (a similarity transformation) and reflection (also a similarity transformation) are applied, the triangles are similar (because similarity transformations preserve shape, just size and orientation, so they are similar).

Answer:

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

The triangles are similar