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

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?  
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

translation to Quadrant II, then a rotation 90 degrees counterclockwise

GPT-4o mini · Answer

To move triangle $ A B C $ in Quadrant IV to triangle $ E G H $ in Quadrant II and demonstrate that they are congruent, we can analyze the transformations required. Given the coordinates of the triangles: - Triangle $ A B C $: - $ A(1, -1) $ - $ B(1, -3) $ - $ C(4, -3) $ - Triangle $ E G H $: - $ E(-2, 5) $ - $ G(-2, 3) $ - $ H(-5, 3) $ ### Steps to Show Congruence 1. **Reflection**: The first step would involve reflecting triangle $ A B C $ across the y-axis. This will change the coordinates as follows: - $ A(1, -1) $ becomes $ A'(-1, -1) $ - $ B(1, -3) $ becomes $ B'(-1, -3) $ - $ C(4, -3) $ becomes $ C'(-4, -3) $ 2. **Translation**: Notice that after reflection, triangle $ A' B' C' $ is located in Quadrant III. To move it to Quadrant II, we need to translate the triangle to the left and upwards. The translation that achieves this is $ (-1, 6) $: - $ A'(-1, -1) $ translates to $ (-1 - 1, -1 + 6) = (-2, 5) = E $ - $ B'(-1, -3) $ translates to $ (-1 - 1, -3 + 6) = (-2, 3) = G $ - $ C'(-4, -3) $ translates to $ (-4 - 1, -3 + 6) = (-5, 3) = H $ Now the transformed triangle's vertices match exactly with those of triangle $ EGH $. ### Conclusion The required transformations are **reflection to Quadrant III**, then **translation of $ (-1, 6) $**. Thus, the correct answer is: **reflection to Quadrant III, then a translation of $ (-1, 6) $**.