Question

Consider ΔABC
, with vertices A(0,3)
, B(0,1)
, and C(−4,2)
, and ΔDEF
, 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)
Responses

ΔABC
is dilated by a scale factor of 2 and is reflected across the y-axis.
triangle upper A upper B upper C is dilated by a scale factor of 2 and is reflected across the y -axis.

ΔABC
is dilated by a scale factor of 12
and is reflected across the x-axis.
triangle upper A upper B upper C is dilated by a scale factor of Start Fraction 1 over 2 End Fraction and is reflected across the x -axis.

ΔABC
is dilated by a scale factor of −2
and is reflected across the y-axis.
triangle upper A upper B upper C is dilated by a scale factor of negative 2 and is reflected across the y -axis.

ΔABC
is dilated by a scale factor of 14
and is reflected across the y-axis.

Answer 1

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:
1. A dilation by a scale factor of 2 will multiply all of the side lengths of ΔABC by 2.
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:
1. **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**.