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.

Question

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

translation to Quadrant II, then a rotation 90 degrees counterclockwise

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

reflection 90 degrees clockwise, then a translation of (1, 6)

reflection to Quadrant III, then a translation (-1, 6)

Answer 1

To determine the correct sequence of steps that can be used to move triangle \( EGH \) in Quadrant IV to the position of triangle \( ABC \) in Quadrant II, let's analyze their coordinates and the options provided.

1. Identify the coordinates of triangle \( EGH \):

   • \( E(-2, 5) \)
   • \( G(-2, 3) \)
   • \( H(-5, 3) \)

2. Identify the coordinates of triangle \( ABC \):

   • \( A(1, -1) \)
   • \( B(1, -3) \)
   • \( C(4, -3) \)

3. Visualize the movements:

   • The triangle \( EGH \) is located in Quadrant IV, and we want to move it to Quadrant II.

The potential transformations include:

• Reflection: A reflection across an axis changes the sign of the coordinate in the direction of the axis. For example, reflecting across the y-axis would take a point \( (x, y) \) to \( (-x, y) \).

• Translation: A translation simply shifts the position of the triangle without changing its size or orientation.

• Rotation: This would change the orientation of the triangle, which may not be necessary since they appear to be congruent.

Now let’s analyze the provided options:

1. Translation to Quadrant II, then a rotation 90 degrees counterclockwise: This option suggests that the triangle is directly shifted to the second quadrant and then rotated, which could work, but it doesn’t clearly describe moving from IV to II.

2. Reflection to Quadrant I, then a translation of (3, 2): Reflecting in the x-axis or y-axis would move the triangle to Quadrant I, and then moving (3, 2) could be a potential sequence.

3. Reflection 90 degrees clockwise, then a translation of (1, 6): A 90-degree clockwise rotation might not be appropriate for transitioning directly between these two triangles' placements.

4. Reflection to Quadrant III, then a translation (-1, 6): A reflection to Quadrant III from Quadrant IV would change the coordinates, and the subsequent translation may not correctly align with triangle \( ABC \).

Given all this analysis, the most feasible path to congruently align triangles \( EGH \) and \( ABC \) would likely involve reflecting to Quadrant I and adjusting the position with a translation.

Thus, the correct answer is:

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