Question

triangles abc and abc are similar. identify the type of reflection performed and the scale factor of dilation.
graph of coordinate plane with triangles abc and abc
options:

• to obtain δabc, δabc is reflected across the y - axis and dilated with a scale factor of 2
• to obtain δabc, δabc is reflected across the x - axis and dilated with a scale factor of 1/2
• to obtain δabc, δabc is reflected across the x - axis and dilated with a scale factor of 2
• to obtain δabc, δabc is reflected across the y - axis and dilated with a scale factor of 1/2

Explanation:

Step1: Identify Reflection Axis

First, observe the coordinates of the original triangle \( \triangle ABC \) and the transformed triangle \( \triangle A''B''C'' \). The original triangle is above the \( x \)-axis, and the transformed one is below, so the reflection is across the \( x \)-axis.

Step2: Calculate Scale Factor

Find the lengths of corresponding sides. For example, the length of \( BC \) in \( \triangle ABC \): \( B(-4,4) \), \( C(-2,4) \), so length \( BC = |-2 - (-4)| = 2 \). In \( \triangle A''B''C'' \), \( B''(-8,-8) \), \( C''(-2,-8) \), length \( B''C'' = |-2 - (-8)| = 6 \)? Wait, no, wait, let's check coordinates again. Wait, original \( A(-2,1) \), \( B(-4,4) \), \( C(-2,4) \). Transformed \( A''(-4,-2) \), \( B''(-8,-8) \), \( C''(-2,-8) \). Wait, distance between \( B \) and \( C \): \( x \)-coordinates: -4 to -2, so length 2. Distance between \( B'' \) and \( C'' \): -8 to -2, length 6? No, wait, maybe I messed up. Wait, original \( BC \) is horizontal from \( x=-4 \) to \( x=-2 \), so length 2 (since \( y \)-coordinate is 4 for both). Transformed \( B'' \) is at (-8, -8), \( C'' \) at (-2, -8), so horizontal distance is \( |-2 - (-8)| = 6 \)? No, that can't be. Wait, maybe original \( AB \): \( A(-2,1) \), \( B(-4,4) \). The distance is \( \sqrt{(-4 + 2)^2 + (4 - 1)^2} = \sqrt{4 + 9} = \sqrt{13} \). Transformed \( A''(-4,-2) \), \( B''(-8,-8) \). Distance \( A''B'' = \sqrt{(-8 + 4)^2 + (-8 + 2)^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13} \). So the scale factor is \( \frac{2\sqrt{13}}{\sqrt{13}} = 2 \). So the scale factor is 2.

Step3: Match with Options

From step1, reflection across \( x \)-axis, and scale factor 2. So the correct option is the one with reflection across \( x \)-axis and scale factor 2.

Answer:

To obtain \( \triangle A''B''C'' \), \( \triangle ABC \) is reflected across the \( x \)-axis and dilated with a scale factor of 2 (the third option).