Question

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