Question

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

1. 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)
2. Where are the vertices of ∆A′B′C′ located? Show your work or explain your steps. (4 points)
3. 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

The answer:

1. The translation rule (x, y) → (x − 4, y + 3) means that every point in the original triangle is moved 4 units to the left and 3 units up to form the new triangle. Therefore, we can describe the translation as a "horizontal shift of 4 units to the left and a vertical shift of 3 units up."

2. To find the vertices of ∆A′B′C′, we apply the translation rule to each vertex of the original triangle:
- A' = (-3, 1) → (-3 - 4, 1 + 3) = (-7, 4)
- B' = (-3, 4) → (-3 - 4, 4 + 3) = (-7, 7)
- C' = (-7, 1) → (-7 - 4, 1 + 3) = (-11, 4)

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

3. No, ∆ABC is not congruent to ∆A″B″C″. A 90° clockwise rotation about the origin changes the orientation of the triangle, so the corresponding vertices of ∆A″B″C″ are not in the same order as those of ∆ABC. Additionally, the side lengths and angles of ∆A″B″C″ are different from those of ∆ABC. Therefore, ∆ABC is not congruent to ∆A″B″C″.