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

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

rotation of __ degrees counterclockwise and translation of (__)

Answer 1

To determine the transformations that demonstrate the congruence of the triangles, we start by noting the positions and shapes of the triangles in the respective quadrants, specifically focusing on the triangle in quadrant 2 and the triangle in quadrant 1.

1. Rotation: To align the triangle in quadrant 2 with the triangle in quadrant 1, we can apply a rotation. The key vertices of the triangle in quadrant 1 are located in quadrant 1 and need to reach the position of the vertices in quadrant 2. A rotation of 90 degrees counterclockwise around a central point (midpoint of a line connecting points of both triangles) should adjust their positions into congruence.

2. Translation: After rotating the triangle, we may need to translate it in order to match the vertices precisely. By "translating," we shift the entire triangle from its rotated position to the exact location of the corresponding vertices of the other triangle. The translation vector would need to be established based on the coordinates of a specific vertex from the triangle in quadrant 1 to the corresponding vertex in quadrant 2.

In summary, the two transformations to demonstrate that the triangles are congruent are:

• Rotation of 90 degrees counterclockwise and
• Translation of (2, 1) (e.g., from the centroid of the rotated triangle to the target centroid).

Thus, the final answer is: Rotation of 90 degrees counterclockwise and translation of (2, 1).