Question

The point P is the reflection of the point (-7, 5) in the line 5x - 3y = 18.

Answer 1

Answer:

(13,-7)

Step-by-step explanation:

$5x-3y=18 \\ \\ -5x+3y=-18 \\ \\ 3y=5x-18 \\ \\ y=\frac{5}{3}x-6$

The slope of the given line is 5/3. Perpendicular lines have slopes that are negative reciprocals of each other, so we need to find the line with slope -3/5 through P.

$y-5=-\frac{3}{5}(x+7) \\ \\ y-5=-\frac{3}{5}x-\frac{21}{5} \\ \\ y=-\frac{3}{5}x+\frac{4}{5}$

Finding where the lines intersect,

$-\frac{3}{5}x+\frac{4}{5}=\frac{5}{3}x-6 \\ \\ -9x+12=25x-90 \\ \\ 34x=102 \\ \\ x=3 \implies y=-1$

The point where the line of reflection and the perpendicular drawn to it through the point being reflected is the midpoint of the preimage and the image.

Let the coordinates of the image be (a,b). Then, by the midpoint formula,

$\frac{-7+a}{2}=3 \implies -7+a=6 \implies a=13 \\ \\ \frac{5+b}{2}=-1 \implies 5+b=-2 \implies b=-7$

Therefore, P has coordinates (13, -7).