Questions: Pentagon ABCDE and pentagon A'B'C'D'E' are shown on the coordinate plane below. Which two transformations are applied to pentagon ABCDE to create A'B'C'D'E'? Translated according to the rule (x, y) -> (x+8, y+2) and reflected across the x-axis Translated according to the rule (x, y) -> (x+2, y+8) and reflected across the y-axis Translated according to the rule (x, y) -> (x+8, y+2) and reflected across the y-axis Translated according to the rule (x, y) -> (x+2, y+8) and reflected across the x-axis

Pentagon ABCDE and pentagon A'B'C'D'E' are shown on the coordinate plane below.

Which two transformations are applied to pentagon ABCDE to create A'B'C'D'E'?
Translated according to the rule (x, y) -> (x+8, y+2) and reflected across the x-axis
Translated according to the rule (x, y) -> (x+2, y+8) and reflected across the y-axis
Translated according to the rule (x, y) -> (x+8, y+2) and reflected across the y-axis
Translated according to the rule (x, y) -> (x+2, y+8) and reflected across the x-axis

Transcript text: Pentagon $A B C D E$ and pentagon $A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime}$ are shown on the coordinate plane below. Which two transformations are applied to pentagon $A B C D E$ to create $A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime}$ ? Translated according to the rule $(x, y) \rightarrow(x+8, y+2)$ and reflected across the $x$-axis Translated according to the rule $(x, y) \rightarrow(x+2, y+8)$ and reflected across the $y$-axis Translated according to the rule $(x, y) \rightarrow(x+8, y+2)$ and reflected across the $y$-axis Translated according to the rule $(x, y) \rightarrow(x+2, y+8)$ and reflected across the $x$-axis

Solution

Which two transformations are applied to pentagon $A B C D E$ to create $A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime}$?

Coordinates of $A B C D E$

The coordinates of pentagon $A B C D E$ are $A(-4, 4)$, $B(-6, 4)$, $C(-5, 1)$, $D(-2, 2)$, and $E(-2, 4)$.

Reflecting across the x-axis

If we reflect $A B C D E$ across the x-axis, the new coordinates will be $A'(-4, -4)$, $B'(-6, -4)$, $C'(-5, -1)$, $D'(-2, -2)$, and $E'(-2, -4)$.

Translating according to the rule $(x, y) \rightarrow (x+2, y+8)$

Applying the translation rule to the reflected points: $A''(-4+2, -4+8) = (-2, 4)$, $B''(-6+2, -4+8) = (-4, 4)$, $C''(-5+2, -1+8) = (-3, 7)$, $D''(-2+2, -2+8) = (0, 6)$, $E''(-2+2, -4+8) = (0, 4)$. These are not the coordinates of $A'B'C'D'E'$.

Reflecting across the y-axis

If we reflect $A B C D E$ across the y-axis, the new coordinates will be $A'(4, 4)$, $B'(6, 4)$, $C'(5, 1)$, $D'(2, 2)$, and $E'(2, 4)$.

Translating according to the rule $(x, y) \rightarrow (x-8, y-6)$

Applying the translation rule $(x-8, y-6)$ to the points $A, B, C, D, E$: $A'(-4-8, 4-6) = A'(-12, -2)$, $B'(-6-8, 4-6) = B'(-14, -2)$, $C'(-5-8, 1-6) = C'(-13, -5)$, $D'(-2-8, 2-6) = D'(-10, -4)$, and $E'(-2-8, 4-6) = E'(-10, -2)$.

Reflecting across the x-axis, then translate by $(x, y) \rightarrow (x+2, y+8)$

Reflecting gives $A'(-4, -4)$, $B'(-6, -4)$, $C'(-5, -1)$, $D'(-2, -2)$, $E'(-2, -4)$. Translating gives $A''(-2, 4)$, $B''(-4, 6)$, $C''(-3, 7)$, $D''(0, 6)$, $E''(0, 4)$.

Reflecting across x-axis, then translating according to $(x,y) \rightarrow (x+6, y-6)$

Reflect across x-axis: $A'(-4, -4)$, $B'(-6, -4)$, $C'(-5, -1)$, $D'(-2, -2)$, $E'(-2, -4)$. Translate by $(x+6, y-6)$: $A''(2, -10)$, $B''(0, -10)$, $C''(1, -7)$, $D''(4, -8)$, $E''(4, -10)$.

Final Answer

The pentagon is rotated $180^\circ$ about the origin and then translated. The coordinates of $A'B'C'D'E'$ are $A'(4, -7)$, $B'(6, -6)$, $C'(5, -3)$, $D'(2, -4)$, and $E'(2, -6)$. This corresponds to a rotation of $180^\circ$ around the origin followed by a translation.

Translated according to the rule $(x, y) \rightarrow (x+2, y+8)$ and reflected across the $x$-axis.

Translated according to the rule $(x, y) \rightarrow (x+2, y+8)$ and reflected across the $x$-axis

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