Graph ΔGHJ with vertices at G(0, 1), H(4, 0), and J(4, 1). Then graph the image of the triangle after a translation of 3 units up followed by a reflection over the y-axis. Find the lengths of each side of the preimage and the image. Then determine if the two figures are congruent.

Model with Mathematics Create a design in the space at the right, using a series of transformations that produce congruent figures. Exchange designs with a classmate and determine what transformations were used to create their design.

Persevere with Problems Triangle A'B'C' has vertices A'(-4, 5), B'(-1, 4), and C'(-2, 0). Triangle ABC was rotated 90° in a clockwise direction about the origin, translated 2 units up, and reflected over the y-axis. What were the coordinates of the vertices of triangle ABC?

Persevere with Problems Line segment XY has endpoints at X(3, 1) and Y(-2, 0). Its image after a series of transformations has endpoints at X'(0, 1) and Y'(5, 0). Find the series of transformations that maps \(\overline{\text{XY}}\) onto \(\overline{\text{X'Y'}}\). Then find the exact length of both segments.

Justify Conclusions A line segment has endpoints at (a, b) and (c, d). Determine whether the following statements are true or false. Justify your reasoning.

a. The line segment with endpoints at (a + x, b) and (c + x, d) is congruent to the original segment.

b. The line segment with endpoints at \((\cfrac{2}{3}\text{a}, \cfrac{2}{3}\text{b})\) and \((\cfrac{2}{3}\text{c}, \cfrac{2}{3}\text{d})\) is congruent to the original segment.