P = (-8,9) P' = -6,9

Question

R = (-11,1) R' = ?
Triangle PQR is reflected over a vertical line of reflection to create triangle P′Q′R′. What are the coordinates of point R′?

Answer 1

To find the coordinates of point $R^{\prime }$ after reflecting point $R$ over a vertical line of reflection, we first need to determine the vertical line of reflection based on point $P$ and its reflected point $P^{\prime }$.

The coordinates of point $P$ are $(-8,9)$, and its reflection $P^{\prime }$ is $(-6,9)$.

The vertical line of reflection is placed halfway between the x-coordinates of $P$ and $P^{\prime }$. We can calculate the x-coordinate of the line of reflection as follows:

$$\text{Line of reflection}=\frac{x_P+x_{P^{\prime }}}{2}=\frac{-8+(-6)}{2}=\frac{-14}{2}=-7$$

Now that we know the line of reflection is $x=-7$, we can reflect point $R$ located at $(-11,1)$ over this line.

To reflect a point over a vertical line, we can find the distance from the point to the line and then move the same distance on the opposite side of the line. The original x-coordinate of $R$ is $-11$, and the x-coordinate of the line is $-7$.

The distance from $R$ to the line is:

$$\text{Distance}=-11-(-7)=-11+7=-4$$

Now, we reflect point $R$ across the line by moving 4 units to the right (towards the positive x-axis):

$$x_{R^{\prime }}=-7+4=-3$$

The y-coordinate remains unchanged during reflection across a vertical line. Thus, the y-coordinate of point $R^{\prime }$ will still be $1$.

So, the coordinates of point $R^{\prime }$ after the reflection are:

$$R^{\prime }=(-3,1)$$

Therefore, the coordinates of point $R^{\prime }$ are $\boxed{{\displaystyle (-3,1)}}$.