Triangle ABC has vertices A(−3, 1), B(−3, 4), and C(−7, 1).

Question

Part A: If ∆ABC is translated according to the rule (x, y) → (x − 4, y + 3) to form ∆A'B'C', how is the translation described with words? (3 points)
Part B: Where are the vertices of ∆A'B'C' located? Show your work or explain your steps. (4 points)
Part C: Triangle A'B'C' is rotated 90° clockwise about the origin to form ∆A"B"C". Is ∆ABC congruent to ∆A"B"C"? Give details to support your answer. (3 points)

Answer 1

Part A: The translation of ∆ABC according to the rule (x, y) → (x − 4, y + 3) can be described as a translation 4 units to the left and 3 units up.

Part B:
Vertex A (-3, 1) translates to A' (-3 - 4, 1 + 3) = A' (-7, 4)
Vertex B (-3, 4) translates to B' (-3 - 4, 4 + 3) = B' (-7, 7)
Vertex C (-7, 1) translates to C' (-7 - 4, 1 + 3) = C' (-11, 4)

Therefore, the vertices of ∆A'B'C' are A'(-7, 4), B'(-7, 7), and C'(-11, 4).

Part C:
To determine if ∆ABC is congruent to ∆A"B"C", we need to compare the side lengths and angles of the two triangles.

1. Side Lengths:
- The side lengths of ∆ABC are: AB = 3, AC = 4, BC = 4
- The side lengths of ∆A"B"C" are: A"B" = 3, A"C" = 3, B"C" = 3

From the side lengths, we can see that the two triangles do not have the same side lengths and therefore, they are not congruent.

2. Angles:
- The angles of ∆ABC are not preserved in the clockwise rotation, so the angles of ∆A"B"C" are not the same as ∆ABC.

Therefore, ∆ABC is not congruent to ∆A"B"C".