Congruence: Performing and Sequencing Rigid Transformations (CCSS.G-CO.5)
Common Core High School Geometry · Learn by Concept
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Common Core High School Geometry › Congruence: Performing and Sequencing Rigid Transformations (CCSS.G-CO.5)
Point P has coordinates (-2, 1). It is rotated 90° counterclockwise about the origin, then reflected across the y-axis. What are the coordinates of P′ after the transformations?
(-1, -2)
(1, -2)
(-1, 2)
(1, 2)
Explanation
Rotate 90° CCW: use $(x, y) \to (-y, x)$. From $(-2, 1)$ to $(-1, -2)$. Then reflect across the y-axis: $(x, y) \to (-x, y)$, so $(-1, -2)$ goes to $(1, -2)$. If you reverse the order (reflect first, then rotate), you get $(-1, 2)$ instead.
Point P has coordinates (3, -4). It is translated right 5 units and up 2 units, then reflected across the x-axis. What are the coordinates of P′ after the transformations?
(8, 4)
(8, 6)
(-8, -2)
(8, 2)
Explanation
Translate by $(+5, +2)$: $(3, -4) \to (8, -2)$. Reflect across the x-axis: $(x, y) \to (x, -y)$, so $(8, -2) \to (8, 2)$. If you reverse the order (reflect first, then translate), you get $(8, 6)$.
Point P has coordinates (-3, 4). It is reflected across the x-axis, then rotated 90° clockwise about the origin. What are the coordinates of P′ after the transformations?
(-4, 3)
(4, -3)
(4, 3)
(-4, -3)
Explanation
Reflect across the x-axis: $(x, y) \to (x, -y)$, so $(-3, 4) \to (-3, -4)$. Rotate 90° CW: $(x, y) \to (y, -x)$, thus $(-3, -4) \to (-4, 3)$. If you reverse the order (rotate first, then reflect), you get $(4, -3)$.
Point P has coordinates (0, -5). It is translated left 3 units, then reflected across the line y = x. What are the coordinates of P′ after the transformations?
(-8, 0)
(5, 3)
(-5, -3)
(-6, -3)
Explanation
Translate by $(-3, 0)$: $(0, -5) \to (-3, -5)$. Reflect across $y=x$: $(x, y) \to (y, x)$, so $(-3, -5) \to (-5, -3)$. If you reverse the order (reflect first, then translate), you get $(-8, 0)$.
Point P has coordinates (7, -1). It is rotated 270° counterclockwise about the origin, then translated down 4 units. What are the coordinates of P′ after the transformations?
(-1, -11)
(-5, -7)
(-1, -3)
(1, 3)
Explanation
Rotate 270° CCW (same as 90° CW): $(x, y) \to (y, -x)$. From $(7, -1)$ to $(-1, -7)$. Then translate $(0, -4)$: $(-1, -7) \to (-1, -11)$. If you reverse the order (translate first, then rotate), you get $(-5, -7)$.
Point $P$ at $(-2, 1)$ is rotated $90^\circ$ counterclockwise about the origin, then reflected across the $y$-axis. What are the coordinates of $P'$ after the transformations?
$(-1, 2)$
$(-2, -1)$
$(1, -2)$
$(1, 2)$
Explanation
Rotate $(-2, 1)$ by $90^\circ$ CCW: $(x, y) \to (-y, x)$ gives $(-1, -2)$. Reflect across the $y$-axis: $(x, y) \to (-x, y)$ gives $(1, -2)$. Final: $(1, -2)$. If you reverse the order (reflect first, then rotate), you get $(-1, 2)$ (choice A).
Point $P$ at $(3, -4)$ is reflected across the $x$-axis, then translated right 5 units and up 2 units. What are the coordinates of $P'$ after the transformations?
$(8, 6)$
$(8, 2)$
$(2, 6)$
$(7, 6)$
Explanation
Reflect across the $x$-axis: $(3, -4) \to (3, 4)$. Then translate by $(+5, +2)$: $(3, 4) \to (8, 6)$. If you reverse the order (translate first, then reflect), you get $(8, 2)$ (choice B). Using the wrong reflection axis ($y$-axis) leads to $(2, 6)$ (choice C).
Point $P$ at $(4, -1)$ is rotated $90^\circ$ clockwise about the origin, then reflected across the line $y=x$. What are the coordinates of $P'$ after the transformations?
$(4, 1)$
$(-1, 4)$
$(-4, 1)$
$(-4, -1)$
Explanation
Rotate $90^\circ$ CW: $(x, y) \to (y, -x)$, so $(4, -1) \to (-1, -4)$. Reflect across $y=x$: $(x, y) \to (y, x)$, so $(-1, -4) \to (-4, -1)$. If you reverse the order (reflect first, then rotate), you get $(4, 1)$ (choice A).
Point $P$ at $(0, 5)$ is translated left 3 units and down 7 units, then reflected across the $x$-axis. What are the coordinates of $P'$ after the transformations?
$(-3, -12)$
$(-3, 2)$
$(3, -2)$
$(-3, -2)$
Explanation
Translate: $(0, 5) \to (-3, -2)$. Reflect across the $x$-axis: $(x, y) \to (x, -y)$ gives $(-3, 2)$. If you reverse the order (reflect first, then translate), you get $(-3, -12)$ (choice A). Forgetting the reflection leaves $(-3, -2)$ (choice D).
Point $P$ at $(-6, 2)$ is reflected across the line $y=-x$, then rotated $90^\circ$ counterclockwise about the origin. What are the coordinates of $P'$ after the transformations?
$(6, 2)$
$(2, -6)$
$(-6, -2)$
$(-6, 2)$
Explanation
Reflect across $y=-x$: $(x, y) \to (-y, -x)$, so $(-6, 2) \to (-2, 6)$. Rotate $90^\circ$ CCW: $(x, y) \to (-y, x)$, so $(-2, 6) \to (-6, -2)$. If you reverse the order (rotate first, then reflect), you get $(6, 2)$ (choice A). Using the wrong reflection axis (the $x$-axis) yields $(2, -6)$ (choice B).