Question

which transformations could be performed to show that △abc is similar to △a\b\c\?
○ a reflection over the x-axis, then a dilation by a scale factor of 3
○ a reflection over the x-axis, then a dilation by a scale factor of \\(\frac{1}{3}\\)
○ a 180° rotation about the origin, then a dilation by a scale factor of 3
○ a 180° rotation about the origin, then a dilation by a scale factor of \\(\frac{1}{3}\\)

Explanation:

Step1: Analyze Coordinates

First, find coordinates of \( \triangle ABC \) (e.g., \( A(-8, 3) \), \( B(-8, 6) \), \( C(0, 3) \)) and \( \triangle A''B''C'' \) (e.g., \( A''(3, -1) \)? Wait, better: \( C \) is at \( (0, 3) \), \( C'' \) at \( (0, -1) \)? Wait, no, looking at the graph, \( \triangle ABC \) has points: \( A(-8, 3) \), \( B(-8, 6) \), \( C(0, 3) \). \( \triangle A''B''C'' \): \( C''(0, -1) \), \( A''(3, -1) \), \( B''(3, -2) \)? Wait, distance \( AC \) in \( \triangle ABC \): from \( x=-8 \) to \( x=0 \), so length 8. \( A''C'' \): from \( x=0 \) to \( x=3 \)? No, wait, maybe reflection over x-axis first. Reflect \( \triangle ABC \) over x-axis: \( A(-8, -3) \), \( B(-8, -6) \), \( C(0, -3) \). Then dilation: \( A'' \) is at (3, -1)? Wait, no, let's check scale factor. Original \( AC \) length: 8 (from x=-8 to x=0, y same). After reflection, then dilation. If scale factor is \( \frac{1}{3} \), then \( 8 \times \frac{1}{3} \)? No, wait, \( A(-8, 3) \), reflect over x-axis: \( (-8, -3) \). Then dilate by \( \frac{1}{3} \): \( (-8 \times \frac{1}{3}, -3 \times \frac{1}{3}) \approx (-2.666, -1) \)? No, maybe \( C(0, 3) \), reflect over x-axis: (0, -3), then dilate by \( \frac{1}{3} \): (0, -1), which matches \( C''(0, -1) \). \( A(-8, 3) \) reflect: (-8, -3), dilate by \( \frac{1}{3} \): \( (-8/3, -1) \approx (-2.666, -1) \)? Wait, no, the \( A'' \) is at (3, -1)? Wait, maybe I misread coordinates. Wait, the \( \triangle A''B''C'' \) is in the fourth quadrant, near x=3, y=-1. Wait, original \( \triangle ABC \) is in the second quadrant (x negative, y positive). Reflect over x-axis: y becomes negative (fourth quadrant? No, reflection over x-axis: (x, y) → (x, -y), so second quadrant (x negative, y positive) becomes (x negative, y negative) – third quadrant? Wait, no, \( A(-8, 3) \) reflect over x-axis: (-8, -3) (third quadrant). But \( \triangle A''B''C'' \) is in fourth quadrant (x positive, y negative). So maybe 180° rotation: (x, y) → (-x, -y). \( A(-8, 3) \) → (8, -3). Then dilate by \( \frac{1}{3} \): (8/3, -1) ≈ (2.666, -1), close to \( A''(3, -1) \). \( B(-8, 6) \) → (8, -6) → (8/3, -2) ≈ (2.666, -2), close to \( B''(3, -2) \). \( C(0, 3) \) → (0, -3) → (0, -1) (dilate by \( \frac{1}{3} \)). So 180° rotation (to get x positive, y negative) then dilation by \( \frac{1}{3} \). Let's check options: last option is "a 180° rotation about the origin, then a dilation by a scale factor of \( \frac{1}{3} \)". Let's verify lengths. \( AC \) in \( \triangle ABC \): distance from (-8,3) to (0,3) is 8 units. \( A''C'' \): distance from (3, -1) to (0, -1) is 3 units? Wait, no, 8 × \( \frac{1}{3} \) ≈ 2.666, but 3 is close (maybe grid is 1 unit per square). Wait, maybe my coordinate reading is off. Alternatively, reflection over x-axis: (x,y)→(x,-y). \( A(-8,3)→(-8,-3) \), \( B(-8,6)→(-8,-6) \), \( C(0,3)→(0,-3) \). Then dilate by \( \frac{1}{3} \): (-8/3, -1), (-8/3, -2), (0, -1). But \( A'' \) is at (3, -1), which is positive x. So 180° rotation: (-8,3)→(8,-3), then dilate by \( \frac{1}{3} \)→(8/3≈2.666, -1), close to (3, -1). So the correct transformation is 180° rotation then dilation by \( \frac{1}{3} \).

Step2: Evaluate Options

- Option 1: Reflection over x-axis (y negative, x negative) then dilation by 3 (makes triangle bigger, but \( \triangle A''B''C'' \) is smaller). Eliminate.
- Option 2: Reflection over x-axis (x negative, y negative) then dilation by \( \frac{1}{3} \) (x still negative, but \( \triangle A''B''C'' \) has positive x). Eliminate.
- Option 3: 180° rotation (x positive, y negative) then dilation by 3 (triangle bigger, but \( \triangle A''B''C'' \) is smaller). Eliminate.
- Option 4: 180° rotation (x positive, y negative) then dilation by \( \frac{1}{3} \) (makes triangle smaller, matches size and position). Correct.

Answer:

a 180° rotation about the origin, then a dilation by a scale factor of \( \frac{1}{3} \) (the fourth option)