Question

triangle xyz is reflected across the y - axis to form figure xyz.
11 x(1, _____)
12 y(_____, 1)
13 z(6, _____)
triangle xyz is reflected across the x - axis to form figure xyz.
14 x(-1, _____)
15 y(_____, -1)
16 z(-6, _____)
17 what do you notice when you compare the coordinates of figure xyz and figure xyz?

Explanation:

Step1: Recall reflection rules

When reflecting a point $(x,y)$ across the $y -$axis, the rule is $(-x,y)$. When reflecting a point $(x,y)$ across the $x -$axis, the rule is $(x, - y)$.

Step2: Find coordinates for reflection across y - axis

For point $X$: Assume the original coordinates of $X$ are $(- 1,5)$ (by observing the graph). When reflected across the $y -$axis, using the rule $(-x,y)$, $X'$ has coordinates $(1,5)$. So the answer for 11 is $5$.
For point $Y$: Assume the original coordinates of $Y$ are $(-4,1)$. When reflected across the $y -$axis, using the rule $(-x,y)$, $Y'$ has coordinates $(4,1)$. So the answer for 12 is $4$.
For point $Z$: Assume the original coordinates of $Z$ are $(-6,4)$. When reflected across the $y -$axis, using the rule $(-x,y)$, $Z'$ has coordinates $(6,4)$. So the answer for 13 is $4$.

Step3: Find coordinates for reflection across x - axis

For point $X$: Assume the original coordinates of $X$ are $(-1,5)$. When reflected across the $x -$axis, using the rule $(x,-y)$, $X''$ has coordinates $(-1,-5)$. So the answer for 14 is $-5$.
For point $Y$: Assume the original coordinates of $Y$ are $(-4,1)$. When reflected across the $x -$axis, using the rule $(x,-y)$, $Y''$ has coordinates $(-4,-1)$. So the answer for 15 is $-4$.
For point $Z$: Assume the original coordinates of $Z$ are $(-6,4)$. When reflected across the $x -$axis, using the rule $(x,-y)$, $Z''$ has coordinates $(-6,-4)$. So the answer for 16 is $-4$.

Step4: Compare the two - sets of coordinates

When comparing the coordinates of $X'Y'Z'$ and $X''Y''Z''$, we notice that if we first reflect across the $y -$axis and then across the $x -$axis, the net transformation is equivalent to a rotation of $180^{\circ}$ about the origin. The $x$ and $y$ coordinates of the points in $X'Y'Z'$ and $X''Y''Z''$ are opposite in sign.

Answer:

1. $5$
2. $4$
3. $4$
4. $-5$
5. $-4$
6. $-4$
7. The $x$ and $y$ coordinates of the points in $X'Y'Z'$ and $X''Y''Z''$ are opposite in sign, which is equivalent to a $180^{\circ}$ rotation about the origin.