To determine which transformations could be used to show that triangles ΔABC and ΔDEF are similar, we need to analyze the two triangles.
Coordinates of the triangles:
- ΔABC has vertices A(0, 3), B(0, 1), C(−4, 2).
- ΔDEF has vertices D(0, 6), E(0, 2), F(8, 4).
Step 1: Calculate the lengths of the sides of each triangle. For ΔABC:
- AB length = |3 - 1| = 2
- BC length = √[(0 - (-4))^2 + (1 - 2)^2] = √[4^2 + (-1)^2] = √[16 + 1] = √17
- AC length = √[(0 - (-4))^2 + (3 - 2)^2] = √[4^2 + 1^2] = √[16 + 1] = √17
For ΔDEF:
- DE length = |6 - 2| = 4
- EF length = √[(0 - 8)^2 + (2 - 4)^2] = √[(-8)^2 + (-2)^2] = √[64 + 4] = √68 = 2√17
- DF length = √[(0 - 8)^2 + (6 - 4)^2] = √[(-8)^2 + 2^2] = √[64 + 4] = √68 = 2√17
Step 2: Analyze the side lengths to determine if they are proportional: The ratios of corresponding side lengths should be equal for the triangles to be similar.
For AB and DE:
- AB/DE = 2/4 = 1/2
For BC and EF:
- BC/EF = √17/(2√17) = 1/2
For AC and DF:
- AC/DF = √17/(2√17) = 1/2
Since the ratios of the corresponding sides are equal (1/2), the triangles are similar.
Step 3: Assess the transformations: To transform ΔABC to ΔDEF:
- A dilation by a scale factor of 2 will multiply all of the side lengths of ΔABC by 2.
- The triangles lie in different quadrants. To align the positions of the vertices properly, we may need a reflection as well. Let's consider if a reflection across the y-axis is needed or if other transformations suffice.
Based on the analysis of the provided options:
- First response: ΔABC is dilated by a scale factor of 2 and reflected across the y-axis. This transformation will help make the triangles similar since the dilation will scale the size correctly, and the reflection will align them with the appropriate orientation.
Conclusion: The correct response is that ΔABC is dilated by a scale factor of 2 and is reflected across the y-axis.
1 answer