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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

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
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Answer 1

To determine how to move triangle ABC in Quadrant IV to triangle EGH in Quadrant II and show that the triangles are congruent, we need to analyze the positions of the triangles and the steps required to match them.

1. Identify the Positions:

   • Triangle ABC (in Quadrant IV) has vertices: A(1, -1), B(1, -3), C(4, -3).
   • Triangle EGH (in Quadrant II) has vertices: E(-2, 5), G(-2, 3), H(-5, 3).

2. Find a Suitable Transformation Sequence:

   • Option 1: Translation to Quadrant II and then a 90-degree rotation.
   • Option 2: A 90-degree clockwise rotation followed by translation.
   • Option 3: Reflection to Quadrant I and then translation.
   • Option 4: Reflection to Quadrant III and then translation.

3. Approach to Transformation:

   • Reflection to Quadrant III: Triangle ABC can be reflected across the x-axis to move from Quadrant IV to I, and then we can perform a translation to move it to the correct position to match triangle EGH.
   • After reflection, the vertices of triangle ABC would be: A(1, 1), B(1, 3), C(4, 3).
   • A translation of (-3, 2) will move triangle A'(1, 1) to E(-2, 5), thus matching positions.

Therefore, the best sequence of transformations is:

Reflection to Quadrant III, then a translation of (-1, 6).

This matches with one of the provided responses.

So the correct answer is: reflection to Quadrant III, then a translation of (−1, 6).