Question

1. let (m), (n), and (k) be lines in the euclidean plane such that (m) is given by the equation (y = 1), line (n) is given by the equation (y = \sqrt{3}(x - 1) + 1), and line (k) is given by the equation (x = 1). graph all three of these lines on the coordinate plane below. notice that all three lines contain the point ((1, 1)). find the equation of the line (\ell) such that (r_{\ell} = r_k r_n r_m).

Explanation:

⚡ Using what you learned: Compositions of Transformations

Step 1: Analyze the lines and their angles

All three lines intersect at the point \( (1, 1) \). We can determine their angles of inclination relative to the horizontal line \( m \):

• Line \( m \): \( y = 1 \) (horizontal line, angle \( \theta_m = 0^\circ \))
• Line \( n \): \( y = \sqrt{3}(x - 1) + 1 \) (slope \( \tan(\theta_n) = \sqrt{3} \implies \theta_n = 60^\circ \))
• Line \( k \): \( x = 1 \) (vertical line, angle \( \theta_k = 90^\circ \))

Step 2: Simplify the composition of reflections

The composition of reflections in two intersecting lines is a rotation about their intersection point by twice the angle between them:

• \( r_n \circ r_m \) is a rotation about \( (1, 1) \) by \( 2 \times (\theta_n - \theta_m) = 2 \times (60^\circ - 0^\circ) = 120^\circ \) counterclockwise.

Now, we compose this rotation with the reflection \( r_k \):

• The composition of three reflections in lines intersecting at a single point \( (1, 1) \) simplifies to a single reflection \( r_\ell \) in a line \( \ell \) that also passes through \( (1, 1) \).

Using the angle relationship for compositions of reflections:

$$ \theta_\ell = \theta_k - \theta_n + \theta_m $$

$$ \theta_\ell = 90^\circ - 60^\circ + 0^\circ = 30^\circ $$

Step 3: Find the equation of line \(\ell\)

The line \( \ell \) passes through \( (1, 1) \) with an angle of inclination of \( 30^\circ \):

• Slope \( M = \tan(30^\circ) = \frac{\sqrt{3}}{3} \)

Using the point-slope form:

$$ y - 1 = \frac{\sqrt{3}}{3}(x - 1) $$

$$ y = \frac{\sqrt{3}}{3}(x - 1) + 1 $$

Answer:

$$ y = \frac{\sqrt{3}}{3}(x - 1) + 1 $$