The triangle ABC where A(1, 2), B(4, 8), C(6, 8) is reflected in the x-axis to triangle A'B'C'. The triangle A'B'C' is then reflected in the origin to the triangle A''B''C''. Write down the coordinates of A'', B'', C''. Write down a single transformation that maps ABC onto A''B''C''.

We know that,

Rule to find reflection of a point in x-axis :

1. Retain the abscissa i.e. x-coordinate.
2. Change the sign of ordinate i.e. y-coordinate.

∴ Coordinates of
⇒ A(1, 2) on reflection in x-axis becomes A'(1, -2).
⇒ B(4, 8) on reflection in x-axis becomes B'(4, -8).
⇒ C(6, 8) on reflection in x-axis becomes C'(6, -8).

We know that,

Rules to find the reflection of a point in the origin :

1. Change the sign of abscissa i.e. x-coordinate.
2. Change the sign of ordinate i.e. y-coordinate.

∴ Coordinates of
⇒ A'(1, -2) on reflection in origin becomes A''(-1, 2).
⇒ B'(4, -8) on reflection in origin becomes B''(-4, 8).
⇒ C'(6, -8) on reflection in origin becomes C''(-6, 8).

A single transformation that maps A ⇒ A'', B ⇒ B'' and C ⇒ C'' is reflection in y-axis.

Hence, the coordinates of A'', B'' and C'' are (-1, 2), (-4, 8) and (-6, 8) respectively, and a single transformation that maps ABC to A''B''C'' is reflection in y-axis.