Points (4, 0) and (-3, 0) are invariant points under reflection in line L₁; points (0, 5) and (0, -2) are invariant under reflection in line L₂.

(i) Name and write the equations of lines L₁ and L₂.

(ii) Write P', the reflection of P(6, -8) in L₁ and P" the image of P in L₂.

(iii) State a single transformation that maps P' onto P".

We know that,

y-coordinate of points lying on x-axis is zero and x-coordinate of points lying on y-axis is zero.

∴ (4, 0) and (-3, 0) lies on x-axis and (0, 5) and (0, -2) lies on y-axis.

Also,

A point is invariant under reflection in the line on which it lies itself.

Given,

(4, 0) and (-3, 0) are invariant points under reflection in line L₁.

(i) Since, (4, 0) and (-3, 0) lies on x-axis so they are invariant under reflection in x-axis.

∴ L₁ = x-axis

(0, 5) and (0, -2) are invariant points under reflection in line L₂.

Since, (0, 5) and (0, -2) lies on y-axis so they are invariant under reflection in y-axis.

∴ L₂ = y-axis

Hence, L₁(x-axis) : y = 0 and L₂(y-axis) : x = 0.

(ii) We know that,

On reflection in x-axis, the sign of y-coordinate changes.

∴ P(6, -8) on reflection in L₁ becomes P'(6, 8).

On reflection in y-axis, the sign of x-coordinate changes.

∴ P(6, -8) on reflection in L₂ becomes P"(-6, -8).

Hence, coordinates of P' = (6, 8) and P" = (-6, -8).

(iii) We know that,

On reflection in origin, the sign of both x-coordinate and y-coordinate changes.

∴ P'(6, 8) on reflection in origin, becomes P"(-6, -8).

Hence, single transformation that maps P' onto P" is reflection in origin.