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Coordinate geometry and transformations - Mathematics Year 9

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Introduction

Which information describes a position, and which describes a movement? Use signed coordinate differences, midpoints and fully specified transformations to answer shape questions exactly.

Complete five coordinate tasks and describe three transformations. Keep labelled correspondences, centres or mirror lines and the transformation order, then check lengths and orientation without measuring the screen.

Solve five coordinate problems with signed differences

A coordinate pair(x, y) names a position relative to the chosen origin and axes. A displacement pair names a change in position: horizontal change first, vertical change second. Their two-number notation can look alike, but the mathematical roles differ. The vector(4, −2) means move four units right and two down, not “go to point 4, −2”.

For the source rectangle A=(1, 1), B=(5, 1), C=(5, 3), D=(1, 3), horizontal sides have length|5−1|=4 and vertical sides|3−1|=2 units. The perimeter is 2(4+2)=12 units and area 4×2=8 square units. Side length is nonnegative even if subtracting coordinates in the reverse order gives a negative displacement.

Five complete coordinate tasks.

Task	Given data	Required result	Reason or check
1	Rectangle A 1, 1 B 5, 1 C 5, 3 D 1, 3	Width 4, height 2	5−1=4 and 3−1=2
2	A 1, 1 B 5, 1 C 5, 3; complete the rectangle	D 1, 3	D shares x with A and y with C
3	Diagonal endpoints A 1, 1 and C 5, 3	Midpoint 3, 2	Average x 3, y 2
4	Endpoints −3, 5 and 5, −1	Midpoint 1, 2	Average x 1, y 2
5	Point −1, 3; translation 4, −2	Image 3, 1	−1+4=3; 3−2=1

The missing vertex in task 2 has x 1 from the left side and y 3 from the upper side, so D=(1, 3). Check all four sides remain horizontal or vertical and opposite lengths match. This shortcut depends on the given axis-aligned rectangle; it is not a rule that an arbitrary three-point polygon's fourth vertex can be found by copying coordinates without further conditions.

The midpoint of endpoints(x₁, y₁), (x₂, y₂) is $M = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$ . Average the horizontal coordinates separately from the vertical coordinates. For the source diagonal A 1, 1 to C 5, 3, this gives(3, 2). From A to M and M to C, the displacements are both(2, 1), independently checking the halfway property.

In task 4, the endpoints have signed coordinates(−3, 5) and(5, −1). The midpoint is((−3+5)/2, (5−1)/2)=(1, 2). Both endpoint-to-midpoint displacements are(4, −3). Averaging −3 with 5 is not the same as averaging −3 with the other point's y coordinate −1.

In source task 5, translate P=(−1, 3) by(4, −2) to obtain P′=(3, 1). The inverse displacement(−4, 2) returns to P. A translation applies the same displacement to every point, so one checked point supports arithmetic but does not alone establish which transformation was used for a whole figure.

Keep labels, axes and coordinate units clear. A mathematical grid can model a location or a shape, but it does not automatically represent metres or a real survey. Exact differences and midpoint algebra do not require measuring a printed diagram's apparent dimensions.

Describe three transformations precisely

Three precisely described transformations. Start with triangle ABC: A=(−2, 1), B=(1, 1), C=(−2, 3). The table gives each image under a separate transformation of the original triangle; do not apply the rows successively unless explicitly asked to compose them.

Transformation	Complete description	A image	B image	C image
Translation	Vector 4, −2	2, −1	5, −1	2, 1
Reflection	Mirror line x=0	2, 1	−1, 1	2, 3
Rotation	90° clockwise, centre 0, 0	1, 2	1, −1	3, 2

The translation adds 4 to every x and subtracts 2 from every y. For example A becomes(2, −1) and C becomes(2, 1). Each image point differs from its original by exactly(4, −2). Side differences such as B−A=(3, 0) are unchanged because the same translation cancels when subtracting image points.

The reflection has mirror line x=0, the y-axis. It sends(x, y) to(−x, y). Matched points have equal perpendicular distances on opposite sides of that line, and points on the mirror stay fixed. Specify the line: a reflection in the x-axis would instead change the y sign and produce different images.

The 90° clockwise rotation about the origin sends $(x, y) \mapsto (y, - x)$ . Thus A −2, 1 becomes 1, 2; B 1, 1 becomes 1, −1; C −2, 3 becomes 3, 2. A quarter-turn anti-clockwise would use(−y, x), so direction and centre are both necessary. A bare description “rotate 90°” is incomplete.

All three are rigid motions and preserve lengths and angles. A reflection reverses the orientation of an ordered polygon, while a translation or rotation preserves it. “Clockwise rotation” describes the direction of motion; it does not mean that an originally anti-clockwise labelled triangle cycle must become clockwise after rotation.

Compare a complete point correspondence with checking one vertex. Source point P=(−1, 3) is sent to(3, 1) both by translation(4, −2) and by a 90° clockwise origin rotation. These are different transformations despite agreeing at P. A second point Q=(−2, −1) goes to(2, −3) under the translation but(−1, 2) under the rotation, so it distinguishes them.

Retain all three transformed triangles with vertex labels and complete descriptions. Check original and image squared distances rather than estimating the lengths of tilted sides from screen pixels. Record actual plotting accuracy separately if a real drawing is inspected.

Plot, compose and independently check coordinate reasoning

Draw the source coordinate figure. The original schematic below marks the rectangle's vertices and diagonal midpoint M. Each grid interval is one unit. Join A→B→C→D→A in your own plotted diagram, and draw diagonals to check their common midpoint. The dots are blank positions, not additional vertices.

Source rectangle and midpoint
y
3 | .  D  .  .  .  C
2 | .  .  .  M  .  .
1 | .  A  .  .  .  B
0 | O  .  .  .  .  .
  +------------------
    0  1  2  3  4  5 x
Each interval is one unit.

For rectangle ABCD, the second diagonal from B=(5, 1) to D=(1, 3) also has midpoint(3, 2). This independently checks the rectangle's common-diagonal-midpoint property. The diagonal length is√(4²+2²)=√20=2√5 units, not the side sum 6. The Pythagorean step is justified by the horizontal/vertical right angle in the coordinate system.

Order matters in combined transformations. Apply translation T=(4, −2), then reflection R in the y-axis, to P=(−1, 3). First T gives(3, 1), then R gives(−3, 1). Reverse the order: R first gives(1, 3), then T gives(5, 1). The different final points show these transformations do not commute in this example. Write the intermediate point to make the order visible.

The original Corbettmaths rotation task turns a triangle 90° clockwise about the origin. Track a labelled vertex and its centre-relative position before comparing its image. Its triangle is separate from our A −2, 1/B 1, 1/C −2, 3 data; copy the method and complete description, not its coordinates. Original: corbettmaths, Rotations - Corbettmaths. The written explanations and native tasks work without sound or the external player.

Unfamiliar transfer. Translate Q=(2, −4) by(−3, 5) to obtain(−1, 1). The inverse(3, −5) restores(2, −4). A new segment from(−4, −2) to(2, 6) has midpoint(−1, 2): the two half-displacements are(3, 4). Its total displacement is(6, 8) and length 10 units, so each half-length is 5. These provide different position, vector and length checks from the same data.

First-error analysis. Adding the vector entries to one another gives 2, but that is neither the source translation vector nor its image point. Averaging all four endpoint coordinates into one number loses the horizontal/vertical distinction. Applying a reflection rule without naming the mirror line can also produce a plausible wrong answer. Correct the first invalid operation before recalculating.

Keep the five coordinate solutions, three complete transformation descriptions, plotted correspondences, composition order and new-value transfer. Exact coordinate reasoning is evidence of a mathematical method; it does not certify unobserved drawing, measurement or later mastery. Record actual later retrieval separately.

Interactive Tasks

Quiz: Test Your Knowledge

Source rectangle width and height? (4 and 2 units) (!6 and 4 units) (!5 and 3 units) (!4 and 3 units)

Missing rectangle vertex after 1, 1; 5, 1; 5, 3? (1, 3) (!3, 1) (!5, 5) (!1, 5)

Midpoint of 1, 1 and 5, 3? (3, 2) (!2, 3) (!6, 4) (!4, 2)

Midpoint of −3, 5 and 5, −1? (1, 2) (!−4, 3) (!2, 4) (!4, −3)

Translate −1, 3 by 4, −2. (3, 1) (!4, −2) (!−5, 5) (!1, 3)

Reflection in the y-axis sends x, y to? (−x, y) (!x, −y) (!y, x) (!−x, −y)

90° clockwise about the origin sends −2, 1 to? (1, 2) (!−1, −2) (!2, −1) (!−2, −1)

Which rigid motion reverses polygon orientation? (Reflection) (!Translation) (!Rotation) (!Every enlargement has the same effect)

Translate then reflect source P: final point? (−3, 1) (!5, 1) (!3, −1) (!1, −3)

Does one matching point uniquely identify a transformation of a figure? (No; different transformations may agree there) (!Yes; one point always proves it) (!Only if the point is positive) (!Only for a large screen)

Memory Game

Coordinate	Position relative to axes and origin
Displacement	Signed horizontal and vertical change
Midpoint	Point halfway between endpoints
Translation	Same displacement applied to every point
Reflection	Rigid motion across a stated mirror line
Rotation	Rigid turn with a stated angle, direction and centre

Drag and Drop

1, 3	Missing source rectangle vertex
3, 2	Source diagonal midpoint
3, 1	Source translated point
1, 2	Source A after clockwise quarter-turn
−3, 1	Source P after translation then y-axis reflection

Crossword Puzzle

COORDINATE	Position entry relative to an axis
MIDPOINT	Halfway point on a segment
VECTOR	Signed displacement representation
REFLECTION	Motion across a stated mirror line
ROTATION	Turn about a stated centre
TRANSLATION	Same displacement at every point

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

Read signed positions: Distinguish a coordinate pair from a displacement pair.
Find rectangle dimensions: Use coordinate differences and positive lengths.
Check the source translation: Add 4 horizontally and −2 vertically, then reverse.
Specify transformations: Name a mirror line or a rotation centre and direction.

Standard

Complete five coordinate tasks: Retain all results and their separate x/y checks.
Describe and draw three transformations: Use the exact triangle table as three separate images.
Plot rectangle and midpoint: Draw both diagonals and verify their common midpoint.
Compose two motions: Keep the intermediate point and compare both orders.

Advanced

Challenge one-point evidence: Use P and Q to distinguish translation from clockwise rotation.
Check orientation and lengths: Compare rigid-motion side differences and ordered cycles.
Transfer new signed data: Translate 2, −4 by −3, 5 and analyse the new midpoint/length.
Retain actual later evidence: Keep five tasks, three descriptions, first-error repair and a real delayed response.

Learning Check

Five coordinate results: Keep horizontal and vertical roles distinct.
Three triangle transformations: Give descriptions and every matched image vertex.
Midpoint and metric checks: Use both diagonals and exact squared distances.
Composition and one-point ambiguity: Show intermediate points and a distinguishing second point.
Unfamiliar transfer: Check a new vector and midpoint with lengths.
First error and portfolio: Retain required solutions and actual work.

Worked checks and assessment guidance

Source rectangle width 4, height 2; missing D 1, 3; diagonal midpoint 3, 2; midpoint of −3, 5 and 5, −1 is 1, 2; source translation −1, 3 by 4, −2 gives 3, 1. Perimeter 12 and area 8 verify dimensions.
Original A −2, 1/B 1, 1/C −2, 3. Translation 4, −2 gives 2, −1/5, −1/2, 1. Reflection in x 0 gives 2, 1/−1, 1/2, 3.90° clockwise about origin gives 1, 2/1, −1/3, 2. These are separate original-to-image tasks.
Rectangle diagonals A–C and B–D both have midpoint 3, 2. Each diagonal length√20=2√5 from horizontal 4/vertical 2. Rigid motions preserve lengths; reflection reverses ordered orientation, rotation and translation preserve it.
P −1, 3 under T then R goes 3, 1→−3, 1; R then T goes 1, 3→5, 1. Translation and clockwise rotation both map P to 3, 1, but Q −2, −1 goes 2, −3 versus −1, 2, so one point does not identify the figure transformation.
Q 2, −4 plus −3, 5 gives −1, 1; inverse 3, −5 returns it. Segment −4, −2 to 2, 6 has midpoint −1, 2, half-displacements 3, 4, total length 10 and half-length 5.
Do not add vector components together as an image point or average all four coordinates into one number. Keep five coordinate tasks, three exact descriptions, plotted labels, composition order, independent checks and actual later retrieval; no physical plotting accuracy is invented.

Learning evidence

Retain five coordinate tasks, three fully described transformations with matched vertices, a midpoint diagram, independent metric/orientation checks, composition order, first-error correction and unfamiliar transfer.

OER and Wikipedia

Department for Education: England mathematics programme of study, Key Stage 3, Source area: Key stage 3 > Subject content > Algebra - Coordinates; Geometry and measures - Transformations. Fluency, reasoning and problem solving. Year 9 allocation is editorial within KS3. Numerical models, tasks and feedback are original; images retain actual Commons attribution and videos name their original publisher. Reviewed 12 November 2026.

2D Cartesian Coordinates.svg, H Padleckas, Public domain. Unchanged original; caption and learning activities are original.

Translation illustration.svg, Oleg Alexandrov, Public domain. Unchanged original; caption and learning activities are original.