Consider AABC, with vertices A (0, 3), B (0, 1), and C (-4, 2), and ADEF, with vertices D (0, 6), E (0, 2),

Question

and F (8,4). Which transformations could be used to show that the triangles are similar? (1 point)
AABC is dilated by a scale
O factor of ≥ and is reflected across the x-axis.
AABC is dilated by a scale
• factor of 2 and is reflected across the y-axis.
AABC is dilated by a scale
• factor of — 2 and is reflected across the y-axis.
AABC is dilated by a scale
O factor of ½ and is reflected across the y-axis.

Answer 1

To determine which transformations can be used to show that triangles AABC and ADEF are similar, we first need to compare the coordinates of the vertices of each triangle.

Vertices of triangle AABC:

• A (0, 3)
• B (0, 1)
• C (-4, 2)

Vertices of triangle ADEF:

• D (0, 6)
• E (0, 2)
• F (8, 4)

Next, let's calculate the lengths of the sides of both triangles.

Triangle AABC

1. AB: Distance between A and B: \[ AB = |3 - 1| = 2 \]
2. AC: Distance between A and C: \[ AC = \sqrt{(0 - (-4))^2 + (3 - 2)^2} = \sqrt{(4)^2 + (1)^2} = \sqrt{16 + 1} = \sqrt{17} \]
3. BC: Distance between B and C: \[ BC = \sqrt{(0 - (-4))^2 + (1 - 2)^2} = \sqrt{(4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \]

Triangle ADEF

1. DE: Distance between D and E: \[ DE = |6 - 2| = 4 \]
2. DF: Distance between D and F: \[ DF = \sqrt{(0 - 8)^2 + (6 - 4)^2} = \sqrt{(-8)^2 + (2)^2} = \sqrt{64 + 4} = \sqrt{68} \]
3. EF: Distance between E and F: \[ EF = \sqrt{(0 - 8)^2 + (2 - 4)^2} = \sqrt{(-8)^2 + (-2)^2} = \sqrt{64 + 4} = \sqrt{68} \]

Now, we analyze the sides:

• Sides of triangle AABC: \(2, \sqrt{17}, \sqrt{17}\)
• Sides of triangle ADEF: \(4, \sqrt{68}, \sqrt{68}\)

Ratio Analysis

• The ratio of the corresponding sides for similarity must be consistent:
  • Side \(AB\) (2) to side \(DE\) (4): \[ \frac{2}{4} = \frac{1}{2} \]
  • Side \(AC\) (\(\sqrt{17}\)) to side \(DF\) (\(\sqrt{68}\)): \[ \frac{\sqrt{17}}{\sqrt{68}} = \frac{\sqrt{17}}{2\sqrt{17}} = \frac{1}{2} \]
  • Side \(BC\) (\(\sqrt{17}\)) to side \(EF\) (\(\sqrt{68}\)): \[ \frac{\sqrt{17}}{\sqrt{68}} = \frac{\sqrt{17}}{2\sqrt{17}} = \frac{1}{2} \]

Since all ratios are equal to \(\frac{1}{2}\), triangle ADEF is obtained from triangle AABC by dilation with a scale factor of \(\frac{1}{2}\).

Reflection Analysis

Additionally, we need to consider the reflection. The orientation of the triangles might require a reflection (which can be detected visually or by comparing preserved and corresponding points).

Conclusion

The transformations that could be used to show that triangle AABC is similar to triangle ADEF's best choice is: "AABC is dilated by a scale factor of \(\frac{1}{2}\) and is reflected across the y-axis."

Thus from the provided options, this is the correct transformation.