The coordinates of triangle A'B'C' created by the translation of triangle ABC and triangle A''B''C'', created by the reflection of the triangle A'B'C' are;
Coordinates of triangle A'B'C' are; (0, 0), (4, 3), (5, -2)
Coordinates of triangle A''B''C'' are; (0, 0), (4, -3), (5, 2)
What is a translation transformation?; A translation transformation is one in which the image is created by the shifting of all the points on the pre-image by a specified amount or value.
The coordinates of A(-2, 1), B(2, 4), C(3, -1)
The rule for the translation of triangle ABC to triangle A'B'C' is; (x, y) → (x + 2, y - 1)
The coordinates of the vertices of triangle A'B'C' are found as follows;
A(-2, 1) → (x, y) → (x + 2, y - 1) → A'(-2 + 2, 1 - 1) = A'(0, 0)
B(2, 4) → (x, y) → (x + 2, y - 1) → B'(2 + 2, 4 - 1) = B'(4, 3)
C(3, -1) → (x, y) → (x + 2, y - 1) → C'(3 + 2, -1 - 1) = C'(5, -2)
The coordinates of the vertices of triangle A'B'C' are; A'(0, 0), B'(4, 3), C'(5, -2)
The coordinates of the point (x, y) when reflected across the x-axis is the point (x, -y), therefore, the coordinates of the image of the triangle A'B'C' following reflection across the x-axis are;
A'(0, 0) → Reflected across the x-axis → A''(0, 0)
B'(4, 3) → Reflected across the x-axis → B''(4, -3)
C'(5, -2) → Reflected across the x-axis → C''(5, 2)
The coordinates of the triangle A''B''C'' are therefore; A''(0, 0), B''(4, 3), C''(5, 2)