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 transformations needed to show that the triangles in quadrants 1 and 2 are congruent, we can analyze their respective coordinates and positions.

Triangles Described:

1. Triangle in Quadrant 4 (Points A):

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

2. Triangle in Quadrant 2 (Points D):

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

3. Triangle in Quadrant 1 (Points G):

   • \( G(1, 1) \)
   • \( H(3, 1) \)
   • \( I(2, 3) \)

Analysis of Triangle D:

• Side \( e \) joins points \( D(-4, 2) \) and \( E(-2, 2) \)
• Side \( g \) joins points \( D(-4, 2) \) and \( F(-3, 4) \)
• Side \( h \) joins points \( F(-3, 4) \) and \( E(-2, 2) \)

Transformation Steps:

1. Rotation: It looks like triangle GHI can be rotated around the origin or the center/midpoint of a particular side to align with triangle DEF. To find this angle, we examine the coordinates:

   • G(1,1) perhaps needs to map to D(-4,2)
   • H(3,1) aligns to E(-2,2)
   • I(2,3) aligns to F(-3,4)

   Let's assume a rotation of 180 degrees around the origin, but we need to check the mapping explicitly:

   • G(1, 1) to D(-4, 2): This requires rotation.
   • H(3, 1) to E(-2, 2): This also requires rotation.
   • I(2, 3) to F(-3, 4): This stays consistent.

   Assuming all triangles have proper angles and dimensions, the most suitable counterclockwise rotation would occur during specific reflections.

2. Translation: After the rotation, we would likely need to translate the triangle along the x-axis to fit the placements. For instance:

   • Moving from coordinates such that (0,0) positions move.
   • It might require translating from (1,1) to Hit (-4,2).
   • Finding a particular translation can help examine offset distances across both triangles.

In summation:

• Rotation: The triangle in Quadrant 1 may be congruent to the triangle in Quadrant 2 through a rotation of 180 degrees counterclockwise.
• Translation: A translation vector can be taken as from the bottom-left of the first triangle to the base of the second triangle.

Thus, the final answer is:

• Rotation of 180 degrees counterclockwise and translation of (-5, 1).