Ask a New Question
Menu

An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 10 to 10 in increments of 1. Three triangles are plotted on the graph. The triangle upper A upper B upper C has its vertices marked with closed points at upper A left parenthesis 2 comma negative 2 right parenthesis, upper B left parenthesis 2 comma negative 4 right parenthesis, and upper C left parenthesis 5 comma negative 4 right parenthesis. The second triangle, placed in the second quadrant, has its vertices marked with closed points at left parenthesis negative 3 comma 6 right parenthesis, left parenthesis negative 5 comma 6 right parenthesis, and left parenthesis negative 5 comma 3 right parenthesis. The third triangle, placed in the third quadrant, has its vertices marked with closed points at left parenthesis negative 2 comma negative 2 right parenthesis, left parenthesis negative 4 comma negative 2 right parenthesis, and left parenthesis negative 4 comma negative 5 right parenthesis. An arrow goes from the point left parenthesis negative 2 comma negative 2 right parenthesis to left parenthesis negative 3 comma 6 right parenthesis; an arrow goes from the point left parenthesis negative 4 comma negative 2 right parenthesis to left parenthesis negative 5 comma 6 right parenthesis; an arrow goes from left parenthesis negative 4 comma negative 5 right parenthesis to left parenthesis negative 5 comma 3 right parenthesis.

what sequence of transformations is depicted in the figure, proving the congruence of the triangles?

rotation of __ degrees clockwise about the origin, then translation of (__)

1 answer

To determine the sequence of transformations that proves the congruence of the triangles mentioned in the problem, we analyze the positions of the triangles and their transformations.

  1. Identify the triangles:

    • Triangle ABC has vertices A(2, -2), B(2, -4), and C(5, -4).
    • Triangle D has vertices D(-3, 6), E(-5, 6), and F(-5, 3).
    • Triangle G has vertices G(-2, -2), H(-4, -2), and I(-4, -5).

  2. Transformation from Triangle G to Triangle D:

    • An arrow points from the point G(-2, -2) to D(-3, 6).
    • An arrow points from H(-4, -2) to E(-5, 6).
    • An arrow points from I(-4, -5) to F(-5, 3).

  3. Rotation Analysis:

    • To align Triangle G with Triangle D, we notice that there is a consistent geometric shift.
    • The movement from point G(-2, -2) to D(-3, 6) indicates we have to rotate the triangle first. By examining angles and positioning, it suggests a rotation about the origin.

  4. Determine the angle:

    • Triangle G needs to be rotated to fit in the second quadrant. A rotation of 180 degrees would transform the triangle from the third quadrant to the second quadrant, flipping its orientation.

  5. Translation:

    • After rotating Triangle G 180 degrees around the origin, we then translate it. The translation from the new position of G’s vertices to the final positions of D's vertices looks like moving down to the left which is consistent with a vector of (-1, 8) since the lower corner of triangle D starts at (-3, 6) from (-2, -2).

So the complete sequence of transformations that proves the congruence of the triangles is:

rotation of 180 degrees clockwise about the origin, then translation of (-1, 8).