A ΔABC with vertices A(1, 2), B(4, 4) and C(3, 7) is first reflected in the line y = 0 onto ΔA'B'C' and then ΔA'B'C' is reflected in the origin onto ΔA"B"C".

Write down the co-ordinates of :

(i) A', B' and C'

(ii) A", B" and C"

Write down the single transformation that maps Δ ABC directly onto ΔA"B"C".

(i) y = 0 is the equation of x-axis.

We know that,

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

Retain the abscissa i.e. x-coordinate.

Change the sign of ordinate i.e. y-coordinate.

∴ Point A'(1, -2) is the image of A(1, 2) on reflection in x-axis.

∴ Point B'(4, -4) is the image of B(4, 4) on reflection in x-axis.

∴ Point C'(3, -7) is the image of C(3, 7) on reflection in x-axis.

The coordinates of the vertices of ΔA'B'C' are A'(1, -2), B'(4, -4), C'(3, -7).

(ii) We know that,

Rule to find reflection of a point in origin :

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

∴ Point A"(-1, 2) is the image of A'(1, -2) on reflection in origin.

∴ Point B"(-4, 4) is the image of B'(4, -4) on reflection in origin.

∴ Point C"(-3, 7) is the image of C'(3, -7) on reflection in origin.

The coordinates of the vertices of ΔA"B"C" are A"(-1, 2), B"(-4, 4), C"(-3, 7).

(iii) Transformation,

A(1, 2) ⇒ A" (-1, 2)

B(4, 4) ⇒ B"(-4, 4)

C(3, 7) ⇒ C"(-3, 7)

A transformation that changes the sign of the x-coordinate while keeping the y-coordinate the same is a reflection in the y-axis.

The single transformation that maps Δ ABC directly onto ΔA"B"C" is a reflection in the y-axis.