totally stuck can someone help please
P P'(–8, –3) for the glide reflection where the translation is (x, y) (x –2, y –5) and the line of reflection is y = –x. Find the coordinates of P.
To find the coordinates of point P after the glide reflection, we need to perform two transformations: a translation and a reflection.
1. Translation:
The given translation is (x, y) → (x – 2, y – 5).
To apply the translation, subtract 2 from the x-coordinate and subtract 5 from the y-coordinate of point P'.
New coordinates after translation: P'(–8 – 2, –3 – 5) = P'(-10, -8)
2. Reflection:
The line of reflection is y = –x. To reflect a point across this line, we need to find the equation of the perpendicular line that passes through P'.
The slope of the given line is -1. The slope of a line perpendicular to this would be the negative reciprocal, which is 1.
Using the point-slope form, we can write the equation of the perpendicular line passing through P':
y - y1 = m(x - x1)
y - (-8) = 1(x - (-10))
y + 8 = x + 10
y = x + 2
Now we need to find the intersection point of the lines y = x + 2 and y = -x. Solving these equations simultaneously, we get:
x + 2 = -x
2x = -2
x = -1
Substituting x back into one of the equations, we get y = -1 + 2 = 1.
Hence, the intersection point is P(-1, 1).
Therefore, after performing the glide reflection with the given translation and line of reflection, the coordinates of P are (-1, 1).