Reflections of Shapes Worksheet for 7th - 10th Grade | Lesson Planet - Free Printable

Here are the step-by-step solutions for each reflection problem.

1) Reflection across the line $x = 3$

Rule: When reflecting across a vertical line like $x = 3$, the y-coordinates stay the same. The x-coordinate changes so that the point is the same distance from the line $x = 3$ but on the opposite side.
Formula: New $x = 3 - (old\ x - 3)$ or simply count the distance.

* Point A $(2, 4)$:
* Distance from $x=3$: $3 - 2 = 1$ unit to the left.
* Move 1 unit to the right of $x=3$: $3 + 1 = 4$.
* New Point: $(4, 4)$
* Point B $(2, 1)$:
* Distance from $x=3$: $3 - 2 = 1$ unit to the left.
* Move 1 unit to the right of $x=3$: $3 + 1 = 4$.
* New Point: $(4, 1)$
* Point C $(4, 2)$:
* Distance from $x=3$: $4 - 3 = 1$ unit to the right.
* Move 1 unit to the left of $x=3$: $3 - 1 = 2$.
* New Point: $(2, 2)$

Final Coordinates: $(4, 4), (4, 1), (2, 2)$

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2) Reflection across the line $y = 7$

Rule: When reflecting across a horizontal line like $y = 7$, the x-coordinates stay the same. The y-coordinate changes based on the distance to the line $y = 7$.

* Point A $(2, 8)$:
* Distance from $y=7$: $8 - 7 = 1$ unit above.
* Move 1 unit below $y=7$: $7 - 1 = 6$.
* New Point: $(2, 6)$
* Point B $(4, 9)$:
* Distance from $y=7$: $9 - 7 = 2$ units above.
* Move 2 units below $y=7$: $7 - 2 = 5$.
* New Point: $(4, 5)$
* Point C $(4, 6)$:
* Distance from $y=7$: $7 - 6 = 1$ unit below.
* Move 1 unit above $y=7$: $7 + 1 = 8$.
* New Point: $(4, 8)$

Final Coordinates: $(2, 6), (4, 5), (4, 8)$

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3) Reflection across $y = 1$

Rule: Reflect across the horizontal line $y = 1$. X stays the same. Calculate new Y.

* Point A $(-2, 2)$:
* Distance from $y=1$: $2 - 1 = 1$ unit above.
* Move 1 unit below $y=1$: $1 - 1 = 0$.
* New Point: $(-2, 0)$
* Point B $(-1, 4)$:
* Distance from $y=1$: $4 - 1 = 3$ units above.
* Move 3 units below $y=1$: $1 - 3 = -2$.
* New Point: $(-1, -2)$
* Point C $(0, 1)$:
* This point is ON the line $y=1$. It does not move.
* New Point: $(0, 1)$
* Point D $(1, 3)$:
* Distance from $y=1$: $3 - 1 = 2$ units above.
* Move 2 units below $y=1$: $1 - 2 = -1$.
* New Point: $(1, -1)$

Final Coordinates: $(-2, 0), (-1, -2), (0, 1), (1, -1)$

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4) Reflection across the x-axis

Rule: Reflecting across the x-axis means flipping up/down. The x-coordinate stays the same, but the sign of the y-coordinate flips (positive becomes negative, negative becomes positive).
Formula: $(x, y) \rightarrow (x, -y)$

* Point A $(1, -1)$: Flip y-sign $\rightarrow$ $(1, 1)$
* Point B $(2, -4)$: Flip y-sign $\rightarrow$ $(2, 4)$
* Point C $(3, -2)$: Flip y-sign $\rightarrow$ $(3, 2)$

Final Coordinates: $(1, 1), (2, 4), (3, 2)$

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5) Reflection across the y-axis

Rule: Reflecting across the y-axis means flipping left/right. The y-coordinate stays the same, but the sign of the x-coordinate flips.
Formula: $(x, y) \rightarrow (-x, y)$

* Point A $(-5, -3)$: Flip x-sign $\rightarrow$ $(5, -3)$
* Point B $(-4, -1)$: Flip x-sign $\rightarrow$ $(4, -1)$
* Point C $(-2, -1)$: Flip x-sign $\rightarrow$ $(2, -1)$
* Point D $(-3, -5)$: Flip x-sign $\rightarrow$ $(3, -5)$

Final Coordinates: $(5, -3), (4, -1), (2, -1), (3, -5)$

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6) Reflection across $x = 2$

Rule: Reflect across the vertical line $x = 2$. Y stays the same. Calculate new X based on distance from 2.

* Point A $(-1, -2)$:
* Distance from $x=2$: $2 - (-1) = 3$ units to the left.
* Move 3 units to the right of $x=2$: $2 + 3 = 5$.
* New Point: $(5, -2)$
* Point B $(-2, -4)$:
* Distance from $x=2$: $2 - (-2) = 4$ units to the left.
* Move 4 units to the right of $x=2$: $2 + 4 = 6$.
* New Point: $(6, -4)$
* Point C $(-1, -5)$:
* Distance from $x=2$: $2 - (-1) = 3$ units to the left.
* Move 3 units to the right of $x=2$: $2 + 3 = 5$.
* New Point: $(5, -5)$
* Point D $(1, -3)$:
* Distance from $x=2$: $2 - 1 = 1$ unit to the left.
* Move 1 unit to the right of $x=2$: $2 + 1 = 3$.
* New Point: $(3, -3)$

Final Coordinates: $(5, -2), (6, -4), (5, -5), (3, -3)$

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Final Answer:
1) $(4, 4), (4, 1), (2, 2)$
2) $(2, 6), (4, 5), (4, 8)$
3) $(-2, 0), (-1, -2), (0, 1), (1, -1)$
4) $(1, 1), (2, 4), (3, 2)$
5) $(5, -3), (4, -1), (2, -1), (3, -5)$
6) $(5, -2), (6, -4), (5, -5), (3, -3)$