"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?

rotation of __ degrees counterclockwise and translation of (__)

Answer 1

To determine the two transformations that demonstrate that the triangles in this scenario are congruent, we first need to analyze their positions and how one can be transformed into the other.

1. Rotation: The triangle in quadrant 1 has vertices at \((1, 1)\), \((3, 1)\), and \((2, 3)\), while the triangle in quadrant 2 has vertices at \((-4, 2)\), \((-2, 2)\), and \((-3, 4)\). To assess the possibility of rotation, we can start by determining the orientation and relative positions of the triangles.

   If we consider rotating the triangle in quadrant 1 around a certain point, we find that rotating around the point \((0, 0)\) by \(180\) degrees counterclockwise would lead to the triangle in quadrant 4 (which has negative \(x\) and \(y\) values moving to positive), while a smaller rotation might demonstrate congruency to the triangle in quadrant 2 due to their aligned shapes. If we look closely, we can analyze that rotating the triangle from quadrant 1 by \(90\) degrees counterclockwise would yield the necessary angles and side lengths.

   Therefore, the first transformation is a rotation of \(90\) degrees counterclockwise.

2. Translation: After the rotation, we can analyze how the triangle from quadrant 1 is placed relative to the triangle in quadrant 2. The last step will involve shifting the triangle from its rotated position to match the vertices of the triangle in quadrant 2. As per the coordinates' alignment, the translation can move those coordinates positioned in quadrant 1 to be aligned similar to quadrant 2.

   The translation required is from their new positions after the rotation to the coordinates of the vertices of the triangle in quadrant 2. Specifically, the point \((1, 1)\) will move to \((-4, 2)\), which indicates moving left by \(5\) units and up by \(1\) unit.

   Therefore, the second transformation is a translation of \((-5, 1)\).

Summary of Transformations:

• Rotation of \(90\) degrees counterclockwise.
• Translation of \((-5, 1)\).

Thus, the full answer is: Rotation of 90 degrees counterclockwise and translation of (-5, 1).