Question

for exercises 10 and 11, use the graph and description to answer the questions. 10. lmn is the image of lmn after a reflection across the line y = 2. what are the coordinates of l, m, and n? 11. error analysis a classmate says that to find the image of a point after it is reflected across the line y = 3, you should keep the x - value the same and add 3 to the y - value. how do you respond to the student?

Explanation:

Step1: Determine the reflection rule

For a reflection across the horizontal line $y = k$, the $x$-coordinate of a point $(x,y)$ remains the same, and the new $y$-coordinate is $y'=2k - y$.

Step2: Assume coordinates of $L$, $M$, $N$

Let's assume $L=(x_1,y_1)$, $M=(x_2,y_2)$, $N=(x_3,y_3)$. After reflection across $y = 2$, the new coordinates are:
For point $L$: $L'=(x_1,2\times2 - y_1)=(x_1,4 - y_1)$
For point $M$: $M'=(x_2,2\times2 - y_2)=(x_2,4 - y_2)$
For point $N$: $N'=(x_3,2\times2 - y_3)=(x_3,4 - y_3)$

Step3: Analyze the error - analysis part

The rule for reflecting a point $(x,y)$ across the line $y = 3$ is that the $x$-value remains the same, and the new $y$-value is $y'=2\times3 - y=6 - y$, not $y + 3$. So the classmate is incorrect.

Answer:

1. The coordinates of $L'$, $M'$, $N'$ are $(x_{L},4 - y_{L})$, $(x_{M},4 - y_{M})$, $(x_{N},4 - y_{N})$ respectively (where $(x_{L},y_{L})$, $(x_{M},y_{M})$, $(x_{N},y_{N})$ are the coordinates of $L$, $M$, $N$).
2. The classmate is incorrect. The correct rule for reflecting a point $(x,y)$ across $y = 3$ is to keep the $x$-value the same and calculate the new $y$-value as $6 - y$, not $y + 3$.