A coordinate plane with 4 quadrants shows x and y axes ranging from negative 5 to 5 in increments of 1. Three triangles are formed by joining three plotted points each. Two of the triangles are joined by connecting their vertices with rays. The coordinates of the plotted points for the triangle in quadrant 4 are left parenthesis 1 comma negative 1 right parenthesis, left parenthesis 1 comma negative 3 right parenthesis, and left parenthesis 3 comma negative 2 right parenthesis. The coordinates for the triangle in quadrant 2 are left parenthesis negative 4 comma 2 right parenthesis, left parenthesis negative 2 comma 2 right parenthesis, and left parenthesis negative 3 comma 4 right parenthesis. The side joining point left parenthesis negative 4 comma 2 right parenthesis and point left parenthesis negative 2 comma 2 right parenthesis is labeled e. The side joining point left parenthesis negative 4 comma 2 right parenthesis and point left parenthesis negative 3 comma 4 right parenthesis is labeled g. The side joining point left parenthesis negative 3 comma 4 right parenthesis and point left parenthesis negative 2 comma 2 right parenthesis is labeled h. The coordinates of the plotted points for the triangle in quadrant 1 are left parenthesis 1 comma 1 right parenthesis, left parenthesis 3 comma 1 right parenthesis, and left parenthesis 2 comma 3 right parenthesis. Three rays join each vertex of the triangle in quadrant 1 to the complementary vertices of the triangle in quadrant 2. A ray connects point left parenthesis 1 comma 1 right parenthesis to point left parenthesis negative 4 comma 2 right parenthesis. A ray connects point left parenthesis 3 comma 1 right parenthesis to point left parenthesis negative 2 comma 2 right parenthesis. A ray connects point left parenthesis 2 comma 3 right parenthesis to point left parenthesis negative 3 comma 4 right parenthesis. The arrow heads are on the vertices of the triangle in quadrant 2.

Question

What two transformations are shown to demonstrate that the triangles are congruent?

(1 point)
rotation of
degrees counterclockwise and translation of (
)

Answer 1

To determine the transformations that demonstrate that the triangles are congruent, we analyze the coordinates of the vertices of the triangles.

Triangle in Quadrant 4 has vertices:

• \( A(1, -1) \)
• \( B(1, -3) \)
• \( C(3, -2) \)

Triangle in Quadrant 2 has vertices:

• \( D(-4, 2) \)
• \( E(-2, 2) \)
• \( F(-3, 4) \)

First, let’s visually check the positions of the triangles. The triangle in Quadrant 2 is located in the second quadrant, and the triangle in Quadrant 4 is in the fourth quadrant, suggesting some rotation.

Now let's analyze the transformations:

1. Rotation: The triangle in Quadrant 4 needs to be rotated around the origin to line up with the triangle in Quadrant 2. Since the vertices need to move from respective locations in Quadrant 4 to Quadrant 2, we'd perform a rotation of 180 degrees counterclockwise about the origin.

2. Translation: After rotating, we need to translate the triangle to align it properly with the triangle in Quadrant 2. By examining the coordinates, the translation can be determined by calculating how far we would need to move the points after rotation.

Thus, the transformations that demonstrate the congruence of the triangles are:

• Rotation of \(180^\circ\) counterclockwise
• Translation of \((-2, 4)\) (This translation moves the centroid of the triangular shape in Quadrant 4 to the centroid of the triangular shape in Quadrant 2.)

Hence, the answer to the question is:

• Rotation of \(180^\circ\) counterclockwise and translation of \((-2, 4)\).