Use the image to answer the question.

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

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

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

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

Answer 1

To determine the sequence of steps used to move triangle ABC in Quadrant IV to triangle EGH in Quadrant II, we can analyze the positions of the vertices.

1. Triangle ABC has its vertices at:

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

2. Triangle EGH has its vertices at:

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

Next, we need to visualize how we can transform triangle ABC into triangle EGH.

• Reflection: If we reflect triangle ABC over the y-axis, the coordinates will transform as follows:

  • A(1, -1) reflects to A'(-1, -1)
  • B(1, -3) reflects to B'(-1, -3)
  • C(4, -3) reflects to C'(-4, -3)

• Translation: After reflection, we can then translate the reflected triangle to match triangle EGH. We can look at the reflected coordinates and determine how to translate them to match the new triangle.

  To translate A'(-1, -1) to E(-2, 5):

  • Move left by 1 (to get x-coordinate from -1 to -2)
  • Move up by 6 (to get y-coordinate from -1 to 5)

Hence, the transformation required is:

• Reflect triangle ABC to Quadrant III.
• Translate it by the vector (-1, 6).

Thus, the answer is:

reflection to Quadrant III, then a translation of (−1, 6).