Question

How far apart are parallel lines m and n such that T0, -123(AXYZ) = (R, Rm) (AXYZ)?
Parallel lines m and n are
units apart.

Answer 1

A reflection about a line gives an image that is at an equal distance from the line but on the opposite (other) side of the line

The distance between the lines m and n is 6 units

Reason:

The given parameters are;

$T_{<0, -12>}\Delta(XYZ) = (R_n \circ R_m) \Delta (XYZ)$

Let Y represent the top vertex of Δ(XYZ), and the coordinates of point Y is (3, 0) we have;

The coordinates of Y following $T_{<0, -12>}$ is Y''(3, -12)

A reflection of  Y across the line m, y = -9, gives the coordinates of the image at Y'(3, -18)

A reflection of the image Y'(3, -18) across the line n, y = -15, gives the image at Y''(3, -12)

The distance between lines m, and line n, is -9 - (-15) = 6

Generally, the difference between two lines of reflection, n, and m, following a composite reflection is 2×(n - m)

Given that the total transformation is -12, we have;

2×(n - m) = -12

Therefore;

$(n - m) = \dfrac{-12}{2} = -6$

Which gives;

m - n = 6

The distance between the two lines = 6 units

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