Reflecting shapes
🎯 In this topic you will
- Reflect shapes in the x-axis or y-axis on a coordinate grid
- Reflect shapes on a coordinate grid given the equation of the mirror line
- Identify a reflection and its mirror line
🧠 Key Words
- congruent
- mirror line
- reflected
- reflect
Show Definitions
- congruent: Shapes that are exactly the same size and shape, even if they are flipped or rotated.
- mirror line: A line that acts as the axis of symmetry in a reflection, where each point is the same distance from the line on opposite sides.
- reflected: The image of a shape after it has been flipped over a mirror line.
- reflect: To flip a shape over a line so that it creates a mirror image.
🔎 Reflection on the x-axis
The diagram shows a coordinate grid.
Triangle A is reflected in the x-axis.
The x-axis is the horizontal axis. This is the mirror line.
The image of triangle A is labelled triangle B.
Triangles A and B are congruent.
❓ EXERCISES
1. Copy each diagram and reflect the shape in the $x$-axis.
Two have been started for you.
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For part a, the triangle is already shown correctly reflected.
For part b, plot each vertex across the $x$-axis to complete the reflection.
For part c, reflect the blue trapezium across the $x$-axis to get its mirror image.
2. In which of these diagrams has shape A been correctly reflected in the $x$-axis?
If the reflection is incorrect, copy the diagram and draw the correct reflection.
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b. Incorrect — the reflection has been drawn across the $y$-axis instead of the $x$-axis.
c. Correct — the green shape has been reflected properly in the $x$-axis.
3. Copy each diagram and reflect the shape in the $y$-axis.
One has been started for you.
👀 Show answer
For part b, reflect the red triangle across the $y$-axis so that its image lies in the positive $x$-region.
For part c, reflect the blue diamond across the $y$-axis, giving a symmetric shape on the left.
🧠 Think like a Mathematician
Task: Zara reflects trapezium ABCD in the $x$-axis. The diagram shows the object, ABCD, and its image, A'B'C'D'.
a) The table shows the coordinates of the vertices of the object and its image. Copy and complete the table.
| Vertex | Object | Image |
|---|---|---|
| A | $(1,\,2)$ | $(1,\,-2)$ |
| B | $(2,\,4)$ | $(2,\,-4)$ |
| C | $(4,\,4)$ | $(4,\,-4)$ |
| D | $(5,\,2)$ | $(5,\,-2)$ |
Zara says: “When you reflect a shape in the $x$-axis, the x-coordinates of the vertices will be the same for the object and the image.”
Sofia says: “When you reflect a shape in the $x$-axis, the y-coordinates of the vertices will be the same for the object and the image.”
b) Are Zara and Sofia correct? Explain why they are correct or incorrect.
👀 Show Answer
- a) Completed table shown above. Rule for reflection in the $x$-axis is $(x,\,y)\,\to\,(x,\,-y)$. The $x$-values stay the same; the $y$-values change sign:
- A $(1,2)\rightarrow(1,-2)$
- B $(2,4)\rightarrow(2,-4)$
- C $(4,4)\rightarrow(4,-4)$
- D $(5,2)\rightarrow(5,-2)$
- b)Zara is correct because reflection in the $x$-axis keeps the $x$-coordinate the same and reverses the sign of the $y$-coordinate. Sofia is incorrect — the $y$-coordinates are not the same; they are negatives of the originals.
❓ EXERCISES
5. In which of these diagrams has shape D been correctly reflected in the $y$-axis?
If the reflection is incorrect, copy the diagram and draw the correct reflection.
👀 Show answer
b. Correct — the green shape is a proper mirror image across the $y$-axis.
c. Incorrect — the blue shape is not reflected across the $y$-axis but shifted instead.
🧠 Think like a Mathematician
Task: Reflect pentagon ABCDE in the $y$-axis. The diagram shows the object, ABCDE, and its image, A'B'C'D'E'.
a) The table shows coordinates of the vertices of the object and its image. Copy and complete the table.
| Vertex | Object | Image |
|---|---|---|
| A | $(-4,\,3)$ | $(4,\,3)$ |
| B | $(-1,\,3)$ | $(1,\,3)$ |
| C | $(-2,\,0)$ | $(2,\,0)$ |
| D | $(-1,\,-2)$ | $(1,\,-2)$ |
| E | $(-4,\,0)$ | $(4,\,0)$ |
b) What can you say about the $x$-coordinates of the vertices of the object and its image?
c) What can you say about the $y$-coordinates of the vertices of the object and its image?
d) Will your answers to parts b and c always be true for whatever shape you reflect in the $y$-axis? Explain.
e) In general, when you reflect a shape in the $x$-axis or the $y$-axis, will the object and the image always be congruent? Explain.
👀 show answer
- a) Completed table above. Rule for reflection in the $y$-axis: $(x,\,y)\rightarrow(-x,\,y)$. So each image point has the same $y$ and the opposite-sign $x$.
- b) The $x$-coordinates are negatives of each other: if the object has $x$, the image has $-x$.
- c) The $y$-coordinates are the same for each corresponding pair.
- d) Yes. For any shape reflected in the $y$-axis, the mapping $(x,\,y)\rightarrow(-x,\,y)$ always holds, so parts b and c are always true.
- e) Yes. Reflections are isometries; they preserve side lengths and angles, so the object and its image are congruent.
❓ EXERCISES
7. The diagram shows eight triangles, labelled $A$ to $H$. Copy and complete these statements. The first one has been done for you.
a. $E$ is a reflection of $A$ in the $y$-axis.
b. $F$ is a reflection of ___ in the ___.
c. $C$ is a reflection of ___ in the ___.
d. $G$ is a reflection of ___ in the ___.
e. $D$ is a reflection of ___ in the ___.
👀 Show answer
b. $F$ is a reflection of $B$ in the $y$-axis.
c. $C$ is a reflection of $H$ in the $y$-axis.
d. $G$ is a reflection of $D$ in the $y$-axis.
e. $D$ is a reflection of $G$ in the $y$-axis.
8. This is part of Marcus’ homework.
a. What are the advantages of using Marcus’ method to reflect the triangle?
b. Are there any disadvantages of using Marcus’ method?
c. Use Marcus’ method to reflect this shape in the $y$-axis.
👀 Show answer
a. Marcus’ method is simple and systematic: by reflecting one vertex at a time across the $y$-axis and plotting points, errors are minimized.
b. It can be time-consuming for complex shapes with many vertices, since each vertex must be plotted individually.
c. The reflected quadrilateral appears symmetrically on the opposite side of the $y$-axis with the same dimensions, but flipped horizontally.
You already know how to reflect a shape when you use the $x$-axis or $y$-axis as the mirror line.
You must also be able to reflect a shape on a coordinate grid in other mirror lines.
To do this, you need to know the equation of the mirror line.
Some examples are shown on the grid on the right.
All vertical lines are parallel to the $y$-axis and have the equation $x = \text{‘a number’}$.
All horizontal lines are parallel to the $x$-axis and have the equation $y = \text{‘a number’}$.
❓ EXERCISES
1. Copy each diagram and reflect the shape in the mirror line with the given equation.
a. mirror line $x = 3$
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b. mirror line $x = 4$
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c. mirror line $x = 2.5$
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d. mirror line $y = 4$
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e. mirror line $y = 3$
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f. mirror line $y = 3.5$
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❓ EXERCISES
2. Copy each diagram and reflect the shape in the mirror line with the given equation.
a. mirror line $x = 4$
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b. mirror line $x = 3$
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c. mirror line $x = 2$
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d. mirror line $y = 3$
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e. mirror line $y = 2$
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f. mirror line $y = 4$
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❓ EXERCISES
3. This is part of Gille’s homework.
a. Explain the mistake Gille has made.
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b. Copy the diagram of shape A and draw the correct reflection.
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❓ EXERCISES
4. The diagram shows shape B on a coordinate grid.
Draw the image of shape B after reflection in the line
a. $x = -1$
b. $y = -2$
c. $x = 0.5$
d. $y = -0.5$
👀 Show answer
Method (general rules). For reflection in a vertical line $x = a$, each point $(x, y)$ maps to $(\,2a - x\,,\, y\,)$. For reflection in a horizontal line $y = b$, each point $(x, y)$ maps to $(\,x\,,\, 2b - y\,)$. Distances to the mirror line are preserved and the image lies on the opposite side.
a. Reflect shape B in $x = -1$ using $(x,y) \to (\,2(-1) - x\,, y\,) = (\,-2 - x\,, y\,)$. The image appears the same distance to the left/right of the line $x=-1$ as the original points are to the other side.
b. Reflect in $y = -2$ using $(x,y) \to (x,\, 2(-2) - y) = (x,\, -4 - y)$. The image is vertically opposite across the line $y=-2$.
c. Reflect in $x = 0.5$ using $(x,y) \to (\,2(0.5) - x\,, y\,) = (\,1 - x\,, y\,)$. The figure flips left–right about the half-unit vertical line.
d. Reflect in $y = -0.5$ using $(x,y) \to (x,\, 2(-0.5) - y) = (x,\, -1 - y)$. The figure flips up–down about the line $y=-0.5$.
5. This is part of Oditi’s homework.
a. Make a copy of this grid.
Use Oditi’s method to draw these reflections.
i. Reflect the triangle in the line $x = 4$
ii. Reflect the parallelogram in the line $y = 5$
iii. Reflect the kite in the line $x = 8$
b. What do you think of Oditi’s method?
Is it easy to follow? Can you think of a better method to use to reflect shapes when the mirror line goes through the shape? Explain your answer.
👀 Show answer
a i. About the vertical line $x = 4$, each vertex $(x, y)$ maps to $(\,2\cdot4 - x\,,\, y\,) = (\,8 - x\,,\, y\,)$. The triangle’s orientation reverses left–right and remains the same height.
a ii. About the horizontal line $y = 5$, map each vertex by $(x, y) \to (x,\, 10 - y)$. The parallelogram flips vertically; bases remain parallel and lengths unchanged.
a iii. About the vertical line $x = 8$, map each vertex by $(x, y) \to (\,16 - x\,,\, y\,)$. The kite appears the same distance to the right/left of $x=8$ with equal edge lengths.
b. Oditi’s approach—reflecting each corner in the mirror line and then joining them—is correct and easy to apply when counting grid squares. However, it can be slow and error‑prone if the mirror line passes through the shape or uses half‑units. A clearer method is to use the coordinate rules: for a vertical line $x=a$, use $(x,y)\to(\,2a-x\,,y)$; for a horizontal line $y=b$, use $(x,y)\to(x,\,2b-y)$. You can still count squares to check that each reflected point is the same perpendicular distance from the mirror line.
🧠 Think like a Mathematician
6. Work independently to answer this investigation.
Context: The diagram shows a rectangle and the line $y = x$.
Method:
- Reflect the rectangle in the line $y = x$ by swapping coordinates: $(x, y) \to (y, x)$.
- Write down at least one alternative method you could use (e.g., perpendicular distances to the line) and compare it with the coordinate-swap method.
- Use your preferred method to reflect each shape in the line $y = x$: i, ii, and iii.
Follow‑up Questions:
👀 Show Answer
- a: Every vertex swaps coordinates across $y = x$. For any point $(x, y)$, the image is $(y, x)$. Join the reflected vertices to get the image rectangle.
- b: Both a coordinate‑swap and a geometric (distance/perpendicular) method are valid. The swap rule is faster and avoids measuring; the geometric construction is helpful when working without coordinates.
- c (i–iii): Reflect each vertex of the given shape using $(x, y) \to (y, x)$; orientations reverse accordingly. Check by ensuring each original point and image are the same perpendicular distance from the line $y = x$.
🧠 Think like a Mathematician
7. Work independently to answer this investigation.
Context: Alicia reflects trapezium $ABCD$ in the line $y = x$. The diagram shows the object $ABCD$ and its image $A'B'C'D'$.
a. Copy and complete the table of coordinates.
| Object | A $(3,\,6)$ | B $(3,\,4)$ | C $(\_\_,\,\_\_)$ | D $(\_\_,\,\_\_)$ |
|---|---|---|---|---|
| Image | A′ $(\_\_,\,\_\_)$ | B′ $(\_\_,\,\_\_)$ | C′ $(\_\_,\,\_\_)$ | D′ $(\_\_,\,\_\_)$ |
b. What do you notice about the coordinates of $ABCD$ and its image $A'B'C'D'$?
c. Write a rule you can use to work out the coordinates of the image of a shape when it is reflected in the line $y = x$.
d. Does your rule in part c work for any shape reflected in the line $y = x$? Explain.
👀 Show Answer
- a: Using the rule for reflection in $y = x$, swap coordinates.
A $(3,6)$ → A′ $(6,3)$; B $(3,4)$ → B′ $(4,3)$.
If C is $(p, q)$, then C′ is $(q, p)$; if D is $(r, s)$, then D′ is $(s, r)$. - b: Each image point has its $x$ and $y$ values swapped compared with the original.
- c: Reflection in $y = x$ maps $(x, y)$ to $(y, x)$.
- d: Yes. Any shape is a set of points; applying $(x, y) \to (y, x)$ to every vertex reflects the whole shape in $y = x$. This rule works universally.
❓ EXERCISES
8. The diagram shows shape $ABCD$ on a coordinate grid. It also shows the line $y = x$.
a. Write the coordinates of the points $A$, $B$, $C$ and $D$.
b. When shape $ABCD$ is reflected in the line $y = x$, the image is $A'B'C'D'$. Use your rule from Question $7$, part $c$ to write the coordinates of the points $A'$, $B'$, $C'$ and $D'$.
c. Copy the diagram. Reflect shape $ABCD$ in the line $y = x$.
d. Check the coordinates of the points $A'$, $B'$, $C'$ and $D'$ you worked out in part $b$ are correct. If any are incorrect, check your answers with a partner.
👀 Show answer
a. Read the coordinates of $A, B, C, D$ directly from the grid (record as ordered pairs).
b. Reflection in $y = x$ swaps each coordinate pair: $(x, y) \to (y, x)$. Therefore $A' = (y_A, x_A)$, $B' = (y_B, x_B)$, $C' = (y_C, x_C)$, $D' = (y_D, x_D)$.
c. Plot each reflected vertex using the swap rule and join them in order to form $A'B'C'D'$.
d. Check that every original point and its image are the same perpendicular distance from the line $y = x$ and that corresponding coordinates are swapped.
9. The diagram shows shape $ABCD$ on a coordinate grid. It also shows the line $y = -x$.
a. Make a copy of the diagram. Reflect $ABCD$ in the line $y = -x$ and label the image $A'B'C'D'$.
b. The table shows the coordinates of the vertices of the object and its image. Copy and complete the table.
| Object | A $(-1,\,2)$ | B $(-1,\,4)$ | C $(\_\_,\,\_\_)$ | D $(\_\_,\,\_\_)$ |
|---|---|---|---|---|
| Image | A′ $(\_\_,\,\_\_)$ | B′ $(\_\_,\,\_\_)$ | C′ $(\_\_,\,\_\_)$ | D′ $(\_\_,\,\_\_)$ |
c. What do you notice about the coordinates of $ABCD$ and its image $A'B'C'D'$?
d. Write a rule you can use to work out the coordinates of the image of a shape when it is reflected in the line $y = -x$.
e. Does your rule in part $d$ work for any shape reflected in the line $y = -x$? Explain your answer.
👀 Show answer
a. Reflect each vertex across $y = -x$.
b. Use the mapping for reflection in $y = -x$: $(x, y) \to (-y, -x)$. Hence A′ $=(-2,\,1)$, B′ $=(-4,\,1)$; if C $(p, q)$, then C′ $=(-q,\,-p)$; if D $(r, s)$, then D′ $=(-s,\,-r)$.
c. Each image point has its coordinates swapped and both signs reversed.
d. Rule: reflection in $y = -x$ sends $(x, y)$ to $(-y, -x)$.
e. Yes. Any shape is a set of points; applying $(x, y) \to (-y, -x)$ to all its vertices reflects the entire figure in $y = -x$.
10. The diagram shows triangle $PQR$ on a coordinate grid. It also shows the line $y = -x$.
a. Write the coordinates of the points $P$, $Q$ and $R$.
b. When shape $PQR$ is reflected in the line $y = -x$, the image is $P'Q'R'$. Use your rule from Question $9$, part $d$ to write the coordinates of the points $P'$, $Q'$ and $R'$.
c. Copy the diagram. Reflect shape $PQR$ in the line $y = -x$.
d. Check the coordinates of the points $P'$, $Q'$ and $R'$ you worked out in part $b$ are correct. If any are incorrect, check your answers with a partner.
👀 Show answer
a. Read the coordinates of $P, Q, R$ from the grid.
b. Use the rule for $y = -x$: $(x, y) \to (-y, -x)$. Thus $P' = (-y_P, -x_P)$, $Q' = (-y_Q, -x_Q)$, $R' = (-y_R, -x_R)$.
c. Plot the reflected vertices and join them to form $P'Q'R'$.
d. Verify each original vertex and its image are symmetric about the line $y = -x$ and that both coordinates are swapped and negated.
❓ EXERCISES
11. The diagram shows shapes $J,\ K,\ L,\ M,\ N,\ P$. Choose the correct equation of the mirror line for each reflection.
a. $J$ and $K$ b. $J$ and $M$ c. $M$ and $N$ d. $K$ and $L$ e. $L$ and $P$
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How to determine each mirror line (text‑only guidance).
- If two shapes are opposite across a vertical line, the mirror is $x = a$ where $a$ is the midpoint of their $x$-coordinates: $a=\dfrac{x_{\text{left}}+x_{\text{right}}}{2}$.
- If they are opposite across a horizontal line, the mirror is $y = b$ with $b=\dfrac{y_{\text{lower}}+y_{\text{upper}}}{2}$.
- If one is the coordinate‑swap of the other (points $(x,y)$ become $(y,x)$), the mirror is $y=x$; if points map to $(-y,-x)$, the mirror is $y=-x$.
Apply these checks on the grid to match each pair (a–e) with the correct given equation option. (If you’d like, I can crop each pair and provide the exact equations.)
12. The diagram shows eight triangles, labelled $A$ to $H$. Identify which of the following are reflections. For each one that is a reflection, write the equation of the mirror line.
a. triangle $A$ to triangle $B$ b. $A$ to $C$ c. $B$ to $F$ d. $B$ to $E$
e. $D$ to $A$ f. $G$ to $E$ g. $C$ to $E$ h. $F$ to $G$ i. $D$ to $H$ j. $E$ to $H$
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Decision rules (use the grid to test each pair):
- Vertical mirror: Corresponding points have the same $y$ and equidistant $x$ values; equation $x=a$.
- Horizontal mirror: Corresponding points have the same $x$ and equidistant $y$ values; equation $y=b$.
- Diagonal $y=x$: Coordinates swap $(x,y)\leftrightarrow(y,x)$.
- Diagonal $y=-x$: Coordinates map $(x,y)\leftrightarrow(-y,-x)$.
Mark a single test vertex for each triangle and check which rule fits; if none, the pair is not a reflection.