Question

pq is reflected across the line x = -1. select all that apply
a pq ≅ pq
b q is plotted at (-4, 4)
c q is plotted at (-4, -4)
d q and q are in the same quadrant
e p and p are in the same quadrant

Explanation:

Step1: Recall reflection property

Reflection is a rigid - motion. Rigid motions preserve distance, so the length of $\overline{PQ}$ is equal to the length of $\overline{P'Q'}$, i.e., $\overline{PQ}\cong\overline{P'Q'}$.

Step2: Find the coordinates of $Q$

From the graph, the coordinates of $Q$ are $(2,4)$.

Step3: Use reflection formula for $x = a$

The formula for reflecting a point $(x,y)$ across the line $x = a$ is $(2a - x,y)$. Here $a=-1$ and $x = 2,y = 4$. Then $2a - x=2\times(-1)-2=-4$ and $y = 4$. So the coordinates of $Q'$ are $(-4,4)$.

Step4: Analyze quadrant positions

The point $Q(2,4)$ is in the first - quadrant and $Q'(-4,4)$ is in the second - quadrant. Let's assume the coordinates of $P$ are $(-1,-2)$ (by observing the graph). After reflection across $x=-1$, the $x$ - coordinate of $P'$ is still $-1$ and the $y$ - coordinate remains the same, so $P$ and $P'$ are in the same quadrant.

Answer:

A. $\overline{PQ}\cong\overline{P'Q'}$
B. $Q'$ is plotted at $(-4,4)$
E. $P'$ and $P$ are in the same quadrant