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Rotating shapes

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visibility 138update 7 months agobookmarkshare

🎯 In this topic you will

Rotate shapes 90° and 180° around a centre of rotation on a coordinate grid
Rotate shapes on a coordinate grid and describe a rotation

🧠 Key Words

anticlockwise
centre of rotation
clockwise

Show Definitions

anticlockwise: A direction of rotation that is opposite to the movement of the hands on a clock.
centre of rotation: The fixed point around which a shape is rotated on a coordinate grid.
clockwise: A direction of rotation that follows the movement of the hands on a clock.

When you rotate a shape, you turn it about a fixed point called the centre of rotation.

You can rotate a shape clockwise or anticlockwise.

You must give the number of degrees by which you are rotating the object. The rotations that are most often used are $90^\circ$ and $180^\circ$. When a shape (the object) is rotated to a new position (the image), the object and the image are always congruent.

📘 Worked example

a. Draw the image of this shape after a rotation of $90^{\circ}$ clockwise about the centre of rotation, which is marked $C$.

b. Draw the image of this shape after a rotation of $180^{\circ}$ about the centre of rotation at $(4,\,3)$.

Answer:

a. Trace or use coordinates relative to $C$. For each vertex with coordinates $(x,\,y)$ and centre $C(h,\,k)$, translate to the centre: $(u,\,v)=(x-h,\,y-k)$. A clockwise rotation of $90^{\circ}$ maps $(u,\,v)\to(v,\,-u)$. Translate back: $(x',\,y')=(h+v,\,k-u)$. Plot the images of all vertices and join them to form the rotated shape.

b. About the centre $(4,\,3)$, a rotation of $180^{\circ}$ uses the rule $(x,\,y)\to(2h-x,\,2k-y)$. Here $(h,\,k)=(4,\,3)$, so $(x',\,y')=(8-x,\,6-y)$. Apply this to each vertex, then connect the images in order.

Why these rules work

Rotate about a centre by first translating the shape so the centre is at the origin, applying the standard rotation matrix, then translating back.
$90^{\circ}$ clockwise: $(u,\,v)\to(v,\,-u)$ ⇒ $(x',\,y')=(h+v,\,k-u)$.
$180^{\circ}$: opposite through the centre ⇒ midpoint at the centre ⇒ $(x',\,y')=(2h-x,\,2k-y)$.
Rotations are isometries: lengths and angles are preserved, so the image is congruent to the object.

❓ EXERCISES

1. Copy each diagram and rotate the shape about the centre, $C$, by the given number of degrees.

$180^\circ$

👀 Show answer

The shape is rotated $180^\circ$ about point $C$, producing a congruent image directly opposite its original position.

$90^\circ$ anticlockwise

👀 Show answer

The rectangle is rotated $90^\circ$ anticlockwise about $C$ so that its longer side becomes vertical.

$90^\circ$ clockwise

👀 Show answer

The triangle is rotated $90^\circ$ clockwise about $C$, moving its base to a vertical position.

$180^\circ$

👀 Show answer

The triangle is rotated $180^\circ$ about $C$, producing a congruent image opposite its starting position.

🧠 Think like a Mathematician

Task: Reflect on Arun’s statements about rotation and congruence.

Arun says:

"When you rotate a shape, the object and the image are always congruent."
"When you rotate a shape through 180°, it doesn’t matter whether you turn the shape clockwise or anticlockwise, as you will end up with the same image."

Activity: Work individually to explain why both of these comments are true. Use diagrams or written reasoning to support your explanation.

Follow-up Questions:

1. Why are a shape and its rotated image always congruent?

2. Why does a 180° rotation give the same result clockwise or anticlockwise?

👀 Show Answer

1: Rotation is a rigid transformation, which preserves side lengths and angles. Therefore, the original shape and the rotated image are congruent.
2: A 180° turn places each point on the opposite side of the center of rotation. Whether you rotate clockwise or anticlockwise, the final position is identical, so the result is the same.

❓ EXERCISES

3. Copy each diagram and rotate the shape about the centre, $C$, by the given number of degrees.

a. $180^{\circ}$

b. $90^{\circ}$ clockwise

c. $180^{\circ}$

d. $90^{\circ}$ anticlockwise

👀 Show answer

Method (use for all parts): For centre $C(h, k)$, translate each vertex to the centre $(u, v)=(x-h,\, y-k)$, apply the rule, then translate back. Rules:

$90^{\circ}$ clockwise: $(u, v)\to(v,\,-u)$ ⇒ $(x', y')=(h+v,\, k-u)$
$90^{\circ}$ anticlockwise: $(u, v)\to(-v,\, u)$ ⇒ $(x', y')=(h-v,\, k+u)$
$180^{\circ}$: $(u, v)\to(-u,\,-v)$ ⇒ $(x', y')=(2h-x,\, 2k-y)$

a. Apply the $180^{\circ}$ rule about $C$ to every vertex; the image sits opposite the object through $C$.

b. Apply the $90^{\circ}$ clockwise rule about $C$.

c. Apply the $180^{\circ}$ rule about $C$.

d. Apply the $90^{\circ}$ anticlockwise rule about $C$.

❓ EXERCISES

4. What is incorrect about Maksim’s solution?

👀 Show answer

Maksim rotated the kite incorrectly. His dotted image is not $90^\circ$ anticlockwise about $C$—it is placed in the wrong position. The correct image should appear on the left of the centre, not on the right.

5. Copy the object onto squared paper and draw the correct image.

👀 Show answer

After a $90^\circ$ anticlockwise rotation about $C$, the kite’s top vertex moves to the left. The dotted outline should be drawn to the left of $C$, maintaining congruence and orientation with the $90^\circ$ turn.

❓ EXERCISES

6. Copy each diagram and rotate the shape, using the information given.

a. $90^{\circ}$ anticlockwise centre $(1,\, 3)$

b. $90^{\circ}$ clockwise centre $(1,\, 1)$

c. $180^{\circ}$ centre $(4,\, 3)$

👀 Show answer

Method: Translate each vertex so the centre is the origin: $(u, v)=(x-h,\, y-k)$. Apply the rotation, then translate back.

a.$90^{\circ}$ anticlockwise about $(1,3)$: $(u, v)\to(-v,\, u)$ ⇒ $(x', y')=(1-v,\, 3+u)$.
b.$90^{\circ}$ clockwise about $(1,1)$: $(u, v)\to(v,\, -u)$ ⇒ $(x', y')=(1+v,\, 1-u)$.
c.$180^{\circ}$ about $(4,3)$: $(u, v)\to(-u,\, -v)$ ⇒ $(x', y')=(8-x,\, 6-y)$.

🧠 Think like a Mathematician

Arun draws rectangle A onto this coordinate grid. Each square on the grid has an area of $1 \text{ cm}^2$.

a) What is the area of rectangle A?

Arun rotates rectangle A 180° about the centre (5, 3) to get rectangle B.

b) What is the area of the combined shape of rectangle A and rectangle B?

Arun rotates rectangle A 90° anticlockwise about the centre (3, 1) to get rectangle C.

c) What is the area of the combined shape of rectangle A and rectangle C?

d) Arun makes this comment:

“Whatever centre of rotation I use, the area of my combined shape will always be the total area of my two rectangles.”

Is Arun correct? Give reasons for your answer.

👀 Show Answer

a) Rectangle A covers 2 squares across by 3 squares high, so its area is $2 \times 3 = 6 \text{ cm}^2$.
b) Rectangle B is congruent to A, also with area $6 \text{ cm}^2$. Since they do not overlap, the combined area is $6 + 6 = 12 \text{ cm}^2$.
c) Rectangle C is also congruent with area $6 \text{ cm}^2$. Here, rectangle A and C overlap partially, so the combined area is less than $12 \text{ cm}^2$. Counting squares, the overlap is $2 \text{ cm}^2$, so the combined area is $6 + 6 - 2 = 10 \text{ cm}^2$.
d) Arun’s statement is not always correct. When shapes overlap, the combined area is less than the total area of the two rectangles. The combined area depends on the amount of overlap.

❓ EXERCISES

8. Draw a coordinate grid that goes from $0$ to $8$ on the $x$-axis and $0$ to $4$ on the $y$-axis.

Draw the rectangle on your grid, which has vertices at $(1,1)$, $(5,1)$, $(5,3)$ and $(1,3)$.

a. What is the area of the rectangle?

b. Rotate the rectangle $180^\circ$ about the centre $(4,2)$ and draw the image.

c. The object and image together make one rectangle. What is the area of this rectangle?

👀 Show answer

a. The rectangle has width $= 5 - 1 = 4$ and height $= 3 - 1 = 2$. Area $= 4 \times 2 = 8\,\text{cm}^2$.

b. After a $180^\circ$ rotation about $(4,2)$, the rectangle maps to a new position symmetric about the point. Coordinates swap accordingly.

c. The object and rotated image form a larger rectangle of width $8$ and height $2$. Area $= 16\,\text{cm}^2$.

9. Make two copies of this diagram.

a. On the first copy, rotate the shape $90^\circ$ clockwise about the centre $(3,3)$.

b. On the second copy, rotate the shape $180^\circ$ about the centre $(4,2)$.

👀 Show answer

a. A $90^\circ$ clockwise rotation about $(3,3)$ moves the shape so that its orientation changes, and the base lies vertically.

b. A $180^\circ$ rotation about $(4,2)$ flips the shape symmetrically across the centre point $(4,2)$.

When you carry out a rotation, or describe a rotation, you need three pieces of information:

the angle of the rotation
the direction of the rotation (clockwise or anticlockwise)
the coordinates of the centre of rotation

📘 Worked example

a. Draw the image of this shape after a rotation of $90^{\circ}$ clockwise about the centre of rotation $(-2,\,-1)$.

b. Describe the rotation that takes shape $A$ to shape $B$.

Answer:

a. Rotate the object $90^{\circ}$ clockwise about the point $(-2,\,-1)$ and redraw the image on the grid.

b. Rotation is $180^{\circ}$. The centre of rotation is $(3,\,3)$.

Method (for part a). Trace the shape, place the pencil point on the centre of rotation, turn the tracing $90^{\circ}$ clockwise, then copy the image onto the grid.

Describing rotations. Always state (1) the angle, (2) the direction — clockwise or anticlockwise — and (3) the coordinates of the centre of rotation. For a $180^{\circ}$ rotation you do not need to say clockwise or anticlockwise; both give the same image.

❓ EXERCISES

1. Copy each diagram and rotate the shape using the given information.

a. $90^\circ$ clockwise, centre $(2,1)$

👀 Show answer

The rectangle rotates $90^\circ$ clockwise about $(2,1)$ to form a horizontal rectangle above the centre of rotation.

b. $90^\circ$ anticlockwise, centre $(-2,2)$

👀 Show answer

The triangle rotates $90^\circ$ anticlockwise about $(-2,2)$ to form a triangle positioned below and to the left of the centre of rotation.

c. $180^\circ$, centre $(-1,-2)$

👀 Show answer

The trapezium rotates $180^\circ$ about $(-1,-2)$ to form the same shape upside down on the opposite side of the centre of rotation.

🧠 Think like a Mathematician

Context: This is how Milosh rotates a shape when the centre of rotation is not on the shape and there is no tracing paper.

Question: Rotate the shape $90^\circ$ clockwise about centre $(1,2)$ using Milosh’s method.

Method (Milosh’s construction):

Draw a vertical line segment from a convenient point on the shape to the centre of rotation.
Rotate this segment $90^\circ$ clockwise about the centre to make a horizontal segment of the same length.
Translate (recreate) the entire shape so that the chosen point now lies at the end of the rotated segment; redraw all vertices using equal distances and right angles.
Repeat or check with another reference point if needed; then outline the final image.

Follow‑up Questions:

a. What do you think of Milosh’s method? Does it make rotating easier? Would it work for all rotations?

b. Use Milosh’s method or your own to rotate each shape $90^\circ$ clockwise about the centre $(2,4)$. Do not use tracing paper.

👀 Show Answer

a: The method is valid because a $90^\circ$ clockwise turn maps vertical segments to horizontal ones of equal length. It is helpful when the centre is off the shape and you cannot trace, but it is most convenient for quarter‑turns. For other angles (e.g., $60^\circ$, $45^\circ$) you would need a protractor or coordinate rules. So it works for $90^\circ$ (and similarly $270^\circ$) easily, and for $180^\circ$ a midpoint construction is simpler.
b: For each vertex $(x,y)$$(h,k)=(2,4)$, you can check your drawing using the coordinate rule for a $90^\circ$ clockwise rotation: translate then map $(u,v)=(x-h,\,y-k) \to (v,-u)$, giving $(x',y')=(h+v,\,k-u)=(2+v,\,4-u)$. Your constructed image should match these positions.

❓ EXERCISES

3a. Copy each diagram and rotate the shape using the given information. Do not use tracing paper.

i. $90^\circ$ clockwise, centre $(2,2)$

ii. $90^\circ$ anticlockwise, centre $(1,1)$

iii. $180^\circ$, centre $(1,0)$

👀 Show answer

i. The L-shape rotates $90^\circ$ clockwise about $(2,2)$. ii. The triangle rotates $90^\circ$ anticlockwise about $(1,1)$. iii. The rectangle rotates $180^\circ$ about $(1,0)$.

3b. Use tracing paper to check your answers to part a.

👀 Show answer

Tracing paper confirms that the rotations in part a are correct.

4. This is part of Rohan’s classwork.

a. What is wrong with Rohan’s answer?

b. Copy the object onto squared paper and draw the correct image.

👀 Show answer

a. Rohan’s answer is wrong because he has not rotated the parallelogram correctly about $(2,3)$ — his dotted line does not preserve the correct distances from the centre of rotation. b. The correct image is the parallelogram rotated $90^\circ$ anticlockwise about $(2,3)$, with all vertices placed at equal rotated positions relative to the centre.

🧠 Think like a Mathematician

Context: This is part of Marcus’s homework. He must describe rotations that take shape $A$ to shape $B$.

Your task: Read Sofia’s and Zara’s feedback to Marcus, then answer the questions a–f.

a. Are Sofia and Zara correct? Explain your answers.

b. In part a of Marcus’s homework, the centre of rotation is at $(1,1)$ because this point is the same on both the object and the image. How can you work out the centre of rotation when no point is shared by both the image and the object (as in part b)?

c. Use your answer to part b to work out the centre of rotation in each diagram. (Both rotations are $180^\circ$.)

d. Complete: “I can find the centre of a $180^\circ$ rotation by _______________.”

e. Does your method for finding the centre of a $180^\circ$ rotation work for a $90^\circ$ rotation? Test your answer on these diagrams.

f. Describe a method you can use to work out the centre of rotation for a $90^\circ$ rotation.

👀 Show Answer

a:Yes. Sofia is right: part a is missing the centre of rotation. Zara is right: for a $180^\circ$ rotation, direction is redundant.
b: Draw segments between two pairs of corresponding vertices (e.g., $AA'$ and $BB'$). Construct the perpendicular bisector of each segment; their intersection is the centre of rotation.
c: For $180^\circ$ rotations the centre is the midpoint of any point and its image. Find the midpoint of $A$ & $A'$ (or $B$ & $B'$) in each diagram — both midpoints coincide at the centre.
d: “I can find the centre of a $180^\circ$ rotation by finding the midpoint of a point and its image (the midpoint of the segment joining any corresponding pair).”
e: No — the midpoint trick is special to $180^\circ$. For a quarter‑turn ($90^\circ$), midpoints are not at the centre. Instead, use the perpendicular‑bisector intersection method (see part b) to locate the centre; check on diagrams (i) and (ii).
f: Method for $90^\circ$: choose two corresponding pairs, draw $AA'$ and $BB'$, construct their perpendicular bisectors, and mark their intersection as the centre. (Equivalently, translate one point to the origin, apply the $90^\circ$ rule to vectors, and back‑solve for the centre.)

❓ EXERCISES

6. The diagram shows seven triangles. Match each rotation with the correct description.

a. A to B

b. B to C

c. C to D

d. C to E

e. F to G

i. $90^\circ$ clockwise, centre $(3,5)$

ii. $180^\circ$, centre $(4,1)$

iii. $180^\circ$, centre $(6,5)$

iv. $90^\circ$ anticlockwise, centre $(3,8)$

v. $180^\circ$, centre $(4,4)$

👀 Show answer

$a \rightarrow i,\quad b \rightarrow iii,\quad c \rightarrow v,\quad d \rightarrow iv,\quad e \rightarrow ii.$

❓ EXERCISES

7. The diagram shows triangles R, S, T, U, V and W on a coordinate grid. Describe the rotation that transforms

a. triangle R to triangle S

b. triangle S to triangle T

c. triangle T to triangle U

d. triangle U to triangle V

e. triangle V to triangle W

👀 Show answer

From the provided image, the centres are difficult to read precisely at this resolution. Based on the grid positions, the rotations appear to be quarter–turns or half–turns about nearby lattice points. If you can share a higher-resolution crop of the first graph, I’ll pinpoint the exact centres and directions for parts a–e immediately.

🧠 Tip

You could use the angle, the direction, or the centre of rotation.

8. The diagram shows seven shapes labelled A to G. Here are seven cards labelled i to vii. Each card shows a rotation of one shape to another. For example, card i means rotate shape A to shape B.

a. Put the cards into groups using one property of the rotations. Describe the property of each group.

b. Sort the cards into different groups using a different property of the rotations. Describe the property of each group.

👀 Show answer

The exact groupings depend on reading the small labels accurately; with this resolution I can’t definitively assign each card. However, two valid grouping strategies are:

By angle: group cards that are $90^\circ$ turns together and cards that are $180^\circ$ turns together.
By direction: among the $90^\circ$ turns, split into clockwise vs anticlockwise.

Share a sharper crop of the A–G grid and I’ll list the precise cards in each group.

9. Describe the rotation that transforms S to T in each diagram.

👀 Show answer

a. $180^\circ$ rotation about the midpoint between S and T: centre $\big(3,\;1.5\big)$.
b. $90^\circ$ anticlockwise rotation about the shared corner: centre $\big(-3,\;2\big)$.
c. $180^\circ$ rotation about the midpoint between S and T: centre $\big(-1,\;3.5\big)$.

📘 What we've learned

Rotations turn a shape about a fixed point (the centre of rotation) by a given angle and direction; distances and angles are preserved (images are congruent).
Direction conventions: anticlockwise is positive; clockwise is negative. Quarter turns are $90^\circ$, half turns are $180^\circ$, and three-quarter turns are $270^\circ$.
About the origin, coordinate rules are: $(x,y)\rightarrow(-y,\;x)$ for $90^\circ$ anticlockwise, $(x,y)\rightarrow(y,\;-x)$ for $90^\circ$ clockwise, $(x,y)\rightarrow(-x,\;-y)$ for $180^\circ$.
About any centre $(a,b)$: translate to the origin, rotate using a rule above, then translate back: $(x,y)\xrightarrow{\text{to origin}}(x-a,\;y-b)\xrightarrow{\text{rotate}}(\dots)\xrightarrow{\text{back}}(\dots)+(a,b)$.
Finding the centre on a grid: for a $180^\circ$ turn it is the midpoint of a point and its image; for a $90^\circ$/$270^\circ$ turn it is where the perpendicular bisectors of a pair of corresponding segments meet.
Quick checks when matching rotations: the orientation of the shape (which way a corner points), the angle turned, and whether the centre lies the same distance from corresponding points.
Equivalences to remember: $270^\circ$ anticlockwise $\equiv 90^\circ$ clockwise, and two quarter turns make a half turn: $90^\circ+90^\circ=180^\circ$.
On coordinate diagrams, label the centre and use right-angle turn marks or the coordinate rules to predict exact images.

Rotating shapes

🎯 In this topic you will

🧠 Key Words

❓ EXERCISES

🧠 Think like a Mathematician

❓ EXERCISES

❓ EXERCISES

❓ EXERCISES

🧠 Think like a Mathematician

❓ EXERCISES

❓ EXERCISES

🧠 Think like a Mathematician

❓ EXERCISES

🧠 Think like a Mathematician

❓ EXERCISES

❓ EXERCISES

🧠 Tip

📘 What we've learned

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