Question

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.What two transformations are shown to demonstrate that the triangles are congruent?(1 point)rotation ofdegrees counterclockwise and translation of ()

Answer 1

To demonstrate that the triangles are congruent through transformations, we need to identify the transformations involved.

1. **Rotation**: The triangle in quadrant 2, formed by the points \((-4, 2)\), \((-2, 2)\), and \((-3, 4)\), can be rotated to align with the triangle in quadrant 1 located at \((1, 1)\), \((3, 1)\), and \((2, 3)\). To find the exact angle, it looks like a \(90\)-degree counterclockwise rotation would work as the points can be visualized aligning properly in those positions.

2. **Translation**: After rotating, the next transformation involved in demonstrating congruence would be a translation. The translation vector can be determined by finding how far the points of the triangle in quadrant 2 need to move to match the corresponding points of the triangle in quadrant 1.

For example:
- Point \((-4, 2)\) needs to move to \((1, 1)\).
- The translation vector will be \( (1 - (-4), 1 - 2) = (5, -1) \).

- Point \((-2, 2)\) needs to move to \((3, 1)\) with the same translation vector calculation, confirming \( (3 - (-2), 1 - 2) = (5, -1) \) again holds.

- Lastly, point \((-3, 4)\) needs to move to point \((2, 3)\):
- Again, it confirms to move \( (2 - (-3), 3 - 4) = (5, -1) \).

Therefore, both transformations are a rotation of \(90\) degrees counterclockwise and a translation of \((5, -1)\).

In conclusion:
**Answer**: rotation of 90 degrees counterclockwise and translation of (5, -1)