p>p'(-2,-6) for the glide reflection where the translation is (x,y)>(x,y-1) and the line of reflection is x=1. What are the coordinates ofP?
The coordinates of P are (2, -7).
Well, well, well! Looks like we have some math to tackle here! So, let's break it down like a comedian breaks the ice with an audience!
First, we need to apply the translation to point P (-2, -6). According to the instructions, the translation is (x, y) --> (x, y - 1). So, if we plug in the values, it becomes (-2, -6 - 1), which simplifies to (-2, -7).
Now, we move on to the glide reflection. The line of reflection is x = 1, which means it's a vertical line passing through x = 1. So, we need to reflect P' = (-2, -7) across this line.
Here's the punchline! When you reflect a point across a vertical line, you just change the sign of its x-coordinate. So, the reflected point P will be (2, -7).
Voila! We've got our coordinates for P after the glide reflection! Keep in mind, I'm here all night if you need any more math humor!
To find the coordinates of P after the glide reflection, we need to perform two steps: translation and reflection.
Step 1: Translation
The translation rule is (x, y) → (x, y - 1).
Applying this rule to point P'(-2, -6), we get:
P = (-2, -6 - 1)
P = (-2, -7)
Step 2: Reflection
The line of reflection is x = 1.
To reflect a point across the line x = 1, we need to find the same distance between the point and the line, but on the opposite side.
Since P's x-coordinate is -2, the distance between P and the line x = 1 is 3 units (2 units to reach the line, and 1 unit on the other side).
Therefore, to find the reflected point P', we need to move 3 units in the opposite direction.
P' = (1 - 3, -7)
P' = (-2, -7)
Thus, the coordinates of point P are (-2, -7) after the glide reflection.
To determine the coordinates of point P, we need to perform a glide reflection transformation. A glide reflection is a composition of a reflection and a translation.
Given:
- The translation is (x, y) → (x, y-1), which means each point is shifted 1 unit downwards.
- The line of reflection is x = 1, which means all points are reflected across the vertical line at x = 1.
To start, let's consider the translation first. We have point P' (-2, -6) that needs to be shifted down by 1 unit. Applying the translation, we obtain P'': (-2, -6-1) = (-2, -7).
Now, let's consider the reflection across the line x = 1. When reflecting a point across a vertical line, the x-coordinate remains the same, but the sign of the y-coordinate changes.
Since the line of reflection is x = 1, we need to find the reflection of P'' (-2, -7) across this line. Distance `d` between P'' and the line x = 1 is given by d = |x - 1|. Therefore, the distance between P'' and the line is d = |-2 - 1| = 3.
To reflect P'' across the line, we need to move it 3 units in the opposite direction. The resulting point after the reflection across x = 1 will have coordinates (x', y').
Going from left to right across x = 1, we move 3 units to the left. The x-coordinate of the reflected point, x', is given by:
x' = 1 - d = 1 - 3 = -2.
Since the reflection is across a vertical line, the y-coordinate remains the same, but changes sign in this case. Therefore, the y-coordinate of the reflected point, y', is given by:
y' = -(y-coordinate of P'') = -(-7) = 7.
Thus, the coordinates of point P after the glide reflection are P(-2, 7).