To determine which transformations can be used to show that the triangles \( \Delta ABC \) and \( \Delta DEF \) are similar, we first need to analyze the properties of both triangles.
Vertices of the Triangles
-
Triangle \( \Delta ABC \):
- \( A(0, 3) \)
- \( B(0, 1) \)
- \( C(-4, 2) \)
-
Triangle \( \Delta DEF \):
- \( D(0, 6) \)
- \( E(0, 2) \)
- \( F(8, 4) \)
Steps to Compare the Triangles
-
Find the lengths of the sides of each triangle.
-
For \( \Delta ABC \):
- Length \( AB = |3 - 1| = 2 \)
- Length \( BC = \sqrt{(0 - (-4))^2 + (1 - 2)^2} = \sqrt{16 + 1} = \sqrt{17} \)
- Length \( AC = \sqrt{(0 - (-4))^2 + (3 - 2)^2} = \sqrt{16 + 1} = \sqrt{17} \)
-
For \( \Delta DEF \):
- Length \( DE = |6 - 2| = 4 \)
- Length \( EF = \sqrt{(0 - 8)^2 + (2 - 4)^2} = \sqrt{64 + 4} = \sqrt{68} \)
- Length \( DF = \sqrt{(0 - 8)^2 + (6 - 4)^2} = \sqrt{64 + 4} = \sqrt{68} \)
-
-
Compare the side lengths to identify a similarity:
- From the above calculations, \( \Delta ABC \) has side lengths of \( 2, \sqrt{17}, \sqrt{17} \).
- \( \Delta DEF \) has side lengths of \( 4, \sqrt{68}, \sqrt{68} \).
- Notice that \( 4 = 2 \times 2 \) and \( \sqrt{68} = \sqrt{17} \times 2 \).
-
Scale Factor:
- The triangles appear similar because their corresponding sides have a proportional relationship with a scale factor of \( 2 \).
-
Reflection:
- To find if a reflection is required, we can notice that \( \Delta ABC \) and \( \Delta DEF \) could potentially be made to overlay by reflecting one triangle over the y-axis, depending on the direction in which angles are arranged.
Possible Transformations
Considering the above findings, the only transformation that makes sense is:
- \( \Delta ABC \) is dilated by a scale factor of \( 2 \) (to match the length of sides to those of \( \Delta DEF \)) and is reflected across the y-axis (to align shapes).
Conclusion
Thus, the correct choice among the given responses is:
ΔABC is dilated by a scale factor of 2 and is reflected across the y-axis.
1 answer