Question

Use the image to answer the question

An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 10 to 10 in increments of 1. Two triangles, upper A upper B upper C and upper E upper G upper H, are plotted on the graph. The triangle upper A upper B upper C has its vertices marked with closed points at upper A left parenthesis 1 comma negative 1 right parenthesis, upper B left parenthesis 1 comma negative 3 right parenthesis, and upper C left parenthesis 4 comma negative 3 right parenthesis. The triangle upper E upper G upper H has its vertices marked with closed points at upper E left parenthesis negative 2 comma 5 right parenthesis, upper G left parenthesis negative 2 comma 3 right parenthesis, and upper H left parenthesis negative 5 comma 3 right parenthesis. In triangle upper E upper G upper H, side upper E upper G is labeled as h, side upper G upper H is labeled as e, and side upper H upper E is labeled as g.

What is the sequence of steps used to move the triangle in Quadrant IV to the triangle in Quadrant II to show that the triangles are congruent?

(1 point)
Responses

reflection to Quadrant III, then a translation of (−1, 6)
reflection to Quadrant III, then a translation of left parenthesis negative 1 comma 6 right parenthesis

translation to Quadrant II, then a rotation 90 degrees counterclockwise
translation to Quadrant II, then a rotation 90 degrees counterclockwise

reflection to Quadrant I, then a translation of (3, 2)
reflection to Quadrant I, then a translation of left parenthesis 3 comma 2 right parenthesis

rotation 90 degrees clockwise, then a translation of (1, 6)
rotation 90 degrees clockwise, then a translation of left parenthesis 1 comma 6 right parenthesis
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Answer 1

To determine the sequence of steps needed to move triangle ABC in Quadrant IV to triangle EGH in Quadrant II and show that they are congruent, we first identify the positions of the triangles.

Triangle ABC has vertices at:
- A(1, -1)
- B(1, -3)
- C(4, -3)

Triangle EGH has vertices at:
- E(-2, 5)
- G(-2, 3)
- H(-5, 3)

To move triangle ABC to the location of triangle EGH, we can perform the following steps:

1. **Reflection**: First, we can reflect triangle ABC across the x-axis. This will change the y-coordinates of the vertices from negative to positive, moving point A from (1, -1) to (1, 1), point B from (1, -3) to (1, 3), and point C from (4, -3) to (4, 3).

2. **Translation**: Then we can translate the reflected triangle \( A'B'C' \) to match the position of triangle EGH. To align the new vertices with those of triangle EGH, we would need to translate by moving left 3 units and up 2 units. This translation can be represented as (-3, 2).

In terms of the provided responses, the correct answer that captures this entire move (including the reflection and translation) is:

**reflection to Quadrant I, then a translation of (3, 2)**.

However, creating the new triangle in the original Quadrant I and then translating it to Quadrant II might be simplified, and it's crucial to recognize that a direct answer matching steps provided in options should be chosen. Hence, the reflection is necessary to position the triangle in the upper half and then translate to match the target triangle in Quadrant II.

Thus, the answer is:
- **reflection to Quadrant I, then a translation of (3, 2)**.