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 in Quadrant IV to the triangle in Quadrant II to show that the triangles are congruent?

(1 point)
Responses

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

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

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

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

Answer 1

To determine the sequence of steps to move triangle ABC in Quadrant IV to triangle EGH in Quadrant II, we should identify the transformations that map the vertices of triangle ABC to those of triangle EGH.

1. Identify the Positioning of the Triangles:

   • Triangle ABC: Vertices A(1, -1), B(1, -3), C(4, -3)
   • Triangle EGH: Vertices E(-2, 5), G(-2, 3), H(-5, 3)

2. Check Reflection:

   • By reflecting triangle ABC across the x-axis, the y-coordinates of points A, B, and C would change signs:
     • A(1, -1) → A(1, 1)
     • B(1, -3) → B(1, 3)
     • C(4, -3) → C(4, 3)

3. Check Translation After Reflection:

   • After reflecting triangle ABC across the x-axis, we can now check how to move it from the position of points A(1, 1), B(1, 3), C(4, 3) to the points of triangle EGH.
   • To move from A(1, 1) to E(-2, 5):
     • The x-coordinate moves from 1 to -2, which is a change of -3.
     • The y-coordinate moves from 1 to 5, which is a change of +4.
   • A similar calculation gives us the translation vector to move all points of the triangle.

To achieve this configuration and notice the transformations:

• A reflection across the x-axis, then we can apply a translation based on the determined changes.

Given the options available, the correct transformation sequence involves reflecting triangle ABC to the appropriate quadrant and then using the resulting translation.

The right choice from the options provided is likely (based on this analysis):

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

This option implies the necessary changes to achieve congruence, based on the reflection across the x-axis and necessary translation to align the triangles.