Congruent Figures Quick Check

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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. Three triangles are plotted on the graph. The first triangle is placed in the first quadrant with vertices marked as closed points at the following coordinates: left parenthesis 3 comma 1 right parenthesis, left parenthesis 3 comma 3 right parenthesis, and left parenthesis 7 comma 1 right parenthesis. The second triangle is placed in the second quadrant with vertices marked as closed points at the following coordinates: left parenthesis negative 1 comma 3 right parenthesis, left parenthesis negative 5 comma 3 right parenthesis, and left parenthesis negative 5 comma 5 right parenthesis. The third triangle is placed in the fourth quadrant with vertices marked as closed points at the following coordinates: left parenthesis 1 comma negative 3 right parenthesis, left parenthesis 1 comma negative 7 right parenthesis, and left parenthesis 3 comma negative 3 right parenthesis. An arrow goes from the point left parenthesis 3 comma 1 right parenthesis to left parenthesis negative 5 comma 3 right parenthesis; an arrow goes from the point left parenthesis 3 comma 3 right parenthesis to left parenthesis negative 5 comma 5 right parenthesis; an arrow goes from left parenthesis 7 comma 1 right parenthesis to left parenthesis negative 1 comma 3 right parenthesis.

Describe 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

rotation, then translation
rotation, then translation

translation, then transformation
translation, then transformation

reflection, then translation
reflection, then translation

translation, then reflection
translation, then reflection
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Answer 1

In order to show that the triangles are congruent by moving the triangle in Quadrant IV to the triangle in Quadrant II, we can analyze the transformation sequence necessary to align the two shapes.

1. Reflection: The triangle in Quadrant IV can be reflected over the line \(y = x\) (or the y-axis) to position it correctly in the second quadrant. This will change the coordinates from the fourth quadrant to the second quadrant.

2. Translation: After reflection, the triangle may need to be translated (shifted) to precisely match the triangle located in the second quadrant.

Given these steps, the correct response is:

reflection, then translation.