Question

Consider triangle ABC, with vertices A(0, 3), B(0, 1), and C(-4, 2), and triangle DEF, with vertices D(0, 6), E(0, 2), and F(8, 4). Which transformations could be used to show that the triangles are similar?

Answer 1

To determine if triangles ABC and DEF are similar, we will first check the corresponding angles and the ratios of their sides.

1. **List the coordinates:**
- Triangle ABC has vertices:
- \( A(0, 3) \)
- \( B(0, 1) \)
- \( C(-4, 2) \)

- Triangle DEF has vertices:
- \( D(0, 6) \)
- \( E(0, 2) \)
- \( F(8, 4) \)

2. **Calculate the lengths of the sides of triangle ABC:**
- **AB:**
\[
AB = \sqrt{(0 - 0)^2 + (3 - 1)^2} = \sqrt{0 + 4} = 2
\]
- **AC:**
\[
AC = \sqrt{(0 - (-4))^2 + (3 - 2)^2} = \sqrt{(4)^2 + (1)^2} = \sqrt{16 + 1} = \sqrt{17}
\]
- **BC:**
\[
BC = \sqrt{(0 - (-4))^2 + (1 - 2)^2} = \sqrt{(4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17}
\]

Thus, the sides of triangle ABC are \( AB = 2 \), \( AC = \sqrt{17} \), and \( BC = \sqrt{17} \).

3. **Calculate the lengths of the sides of triangle DEF:**
- **DE:**
\[
DE = \sqrt{(0 - 0)^2 + (6 - 2)^2} = \sqrt{0 + 16} = 4
\]
- **DF:**
\[
DF = \sqrt{(0 - 8)^2 + (6 - 4)^2} = \sqrt{(8)^2 + (2)^2} = \sqrt{64 + 4} = \sqrt{68} = 2\sqrt{17}
\]
- **EF:**
\[
EF = \sqrt{(8 - 0)^2 + (4 - 2)^2} = \sqrt{(8)^2 + (2)^2} = \sqrt{64 + 4} = \sqrt{68} = 2\sqrt{17}
\]

Thus, the sides of triangle DEF are \( DE = 4 \), \( DF = 2\sqrt{17} \), and \( EF = 2\sqrt{17} \).

4. **Check the ratio of corresponding sides:**
- For side \( AB \) and \( DE \):
\[
\frac{AB}{DE} = \frac{2}{4} = \frac{1}{2}
\]
- For side \( AC \) and \( DF \):
\[
\frac{AC}{DF} = \frac{\sqrt{17}}{2\sqrt{17}} = \frac{1}{2}
\]
- For side \( BC \) and \( EF \):
\[
\frac{BC}{EF} = \frac{\sqrt{17}}{2\sqrt{17}} = \frac{1}{2}
\]

5. **Conclusion about similarity:**
Since the ratios of corresponding sides of triangles ABC and DEF are all equal to \( \frac{1}{2} \), triangle ABC is similar to triangle DEF by the Side-Side-Side (SSS) similarity criterion.

6. **Transformation:**
The transformation from triangle ABC to triangle DEF involves a **scaling (dilation)** with a scale factor of \( 2 \), along with possible **translations** to align the triangles, since both triangles can also be repositioned in the coordinate plane.

Thus, a **scaling transformation** followed by possible **translations** can show that triangles ABC and DEF are similar.