Year 7 Mathematics · AC9M7SP03

Coordinate Transformations

We are learning to describe translations, reflections and rotations of point sets using coordinate rules in the Cartesian plane

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Learning goalsSay it simply

We are learning to describe translations, reflections and rotations of point sets using coordinate rules in the Cartesian plane.

A coordinate transformation maps every point of an object to an image according to one consistent rule. A translation adds the same horizontal and vertical changes to every point, a reflection reverses position across a stated mirror line, and a rotation turns every point through the same angle around a stated centre.

Common origin-centred rules include reflection in the y-axis, which maps (x, y) to (−x, y), reflection in the x-axis, which maps (x, y) to (x, −y), a 90-degree anticlockwise rotation, which maps (x, y) to (−y, x), and a 180-degree rotation, which maps (x, y) to (−x, −y). The rule must be applied to every vertex, not just one convenient point.

Translations, reflections and rotations are rigid transformations, so corresponding lengths, angle measures and area remain invariant and the image is congruent to the original. A reflection reverses orientation, while a translation and rotation preserve cyclic orientation; these properties provide checks on calculated coordinates.

Success criteria

  • I can apply one transformation rule to every vertex of a shape.
  • I can specify the vector, mirror line or centre, angle and direction needed to define a transformation.
  • I can compare an object with its image and identify preserved lengths, angles and area.
Key conceptTeach from the board

Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.

Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.

Example 1

Reflecting a triangle in the y-axis.

  1. Triangle ABC has vertices A(1, 1), B(4, 1) and C(2, 3), and the mirror line is the y-axis.
  2. Apply (x, y) to (−x, y) to A, giving A′(−1, 1), and to B, giving B′(−4, 1).
  3. Apply the same rule to C, giving C′(−2, 3), and join the image vertices in corresponding order.
  4. Check that each point and its image are the same horizontal distance from the y-axis and that corresponding side lengths agree.

Final answer: The reflected triangle has vertices A′(−1, 1), B′(−4, 1) and C′(−2, 3).

Check: The x-coordinates have opposite signs, the y-coordinates are unchanged, and the reflected triangle is congruent with reversed orientation.

Example 2

Rotating a right triangle 90 degrees anticlockwise.

  1. Triangle PQR has vertices P(2, −1), Q(4, −1) and R(4, 2), and it is rotated 90 degrees anticlockwise about the origin.
  2. Apply (x, y) to (−y, x) to P, giving P′(1, 2).
  3. Apply the rule to Q and R, giving Q′(1, 4) and R′(−2, 4).
  4. Compare PQ, which is 2 units, with P′Q′, which is also 2 units, and compare QR, which is 3 units, with Q′R′, which is also 3 units.

Final answer: The rotated triangle has vertices P′(1, 2), Q′(1, 4) and R′(−2, 4).

Check: The image preserves the 2-unit and 3-unit perpendicular sides and therefore preserves shape, area and angle measures.

Example 3

Application problem 1

Problem: Triangle A has vertices (1,1), (4,1), (2,3). Reflect it in the y-axis. Give the image coordinates and state one property preserved by the transformation.

  1. Plan: Represent the information first, then calculate, interpret and independently check the result.
  2. Work: The image vertices are (−1,1), (−4,1), (−2,3). Lengths and angles are preserved, while orientation is reversed.
  3. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Final answer: The image vertices are (−1,1), (−4,1), (−2,3). Lengths and angles are preserved, while orientation is reversed.

Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Application problem 2

Problem: Point P(2, −3) is translated by (4, 1) and then reflected in the x-axis; find the final coordinate and explain why reversing the order gives a different result.

  1. Plan: Write the intermediate coordinate for each order before comparing the final points.
  2. Work: Translation gives (6, −2) and reflection gives (6, 2), while reflecting first gives (2, 3) and then translating gives (6, 4), so the order changes the final y-coordinate.
  3. Interpret: A sequence of transformations must be applied in the stated order because the second rule acts on the first image.

Final answer: Translation gives (6, −2) and reflection gives (6, 2), while reflecting first gives (2, 3) and then translating gives (6, 4), so the order changes the final y-coordinate.

Check: A sequence of transformations must be applied in the stated order because the second rule acts on the first image.

Clean visual examplesOne-page board

Clean one-page examples

AC9M7SP03 - Coordinate Transformations
Example 1

Example 1 Reflecting a triangle in the y-axis. Triangle ABC has vertices A(1, 1), B(4, 1) and C(2, 3), and the mirror line is the y-axis. Apply (x, y) to (−x, y) to A, giving A′(−1, 1), and to B, giving B′(−4, 1). Apply the same rule to C, giving C′(−2, 3), and join the image vertices in corresponding order. Check that each point and its image are the same horizontal distance from the y-axis and that corresponding side lengths agree. Final answer: The reflected triangle has vertices A′(−1, 1), B′(−4, 1) and C′(−2, 3). Check: The x-coordinates have opposite signs, the y-coordinates are unchanged, and the reflected triangle is congruent with reversed orientation.

Example 2

Example 2 Rotating a right triangle 90 degrees anticlockwise. Triangle PQR has vertices P(2, −1), Q(4, −1) and R(4, 2), and it is rotated 90 degrees anticlockwise about the origin. Apply (x, y) to (−y, x) to P, giving P′(1, 2). Apply the rule to Q and R, giving Q′(1, 4) and R′(−2, 4). Compare PQ, which is 2 units, with P′Q′, which is also 2 units, and compare QR, which is 3 units, with Q′R′, which is also 3 units. Final answer: The rotated triangle has vertices P′(1, 2), Q′(1, 4) and R′(−2, 4). Check: The image preserves the 2-unit and 3-unit perpendicular sides and therefore preserves shape, area and angle measures.

Example 3

Example 3 Application problem 1 Problem: Triangle A has vertices (1,1), (4,1), (2,3). Reflect it in the y-axis. Give the image coordinates and state one property preserved by the transformation. Plan: Represent the information first, then calculate, interpret and independently check the result. Work: The image vertices are (−1,1), (−4,1), (−2,3). Lengths and angles are preserved, while orientation is reversed. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills. Final answer: The image vertices are (−1,1), (−4,1), (−2,3). Lengths and angles are preserved, while orientation is reversed. Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Example 4 Application problem 2 Problem: Point P(2, −3) is translated by (4, 1) and then reflected in the x-axis; find the final coordinate and explain why reversing the order gives a different result. Plan: Write the intermediate coordinate for each order before comparing the final points. Work: Translation gives (6, −2) and reflection gives (6, 2), while reflecting first gives (2, 3) and then translating gives (6, 4), so the order changes the final y-coordinate. Interpret: A sequence of transformations must be applied in the stated order because the second rule acts on the first image. Final answer: Translation gives (6, −2) and reflection gives (6, 2), while reflecting first gives (2, 3) and then translating gives (6, 4), so the order changes the final y-coordinate. Check: A sequence of transformations must be applied in the stated order because the second rule acts on the first image.

Curriculum examplesCopied content

Content description: describe transformations of a set of points using coordinates in the Cartesian plane, translations and reflections on an axis, and rotations about a given point.

Questions and answersWith answers

Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.

  1. 1. Translate the point (−2, 5) by the vector (4, −3).

    Check answer

    Answer: The translated point is (2, 2).

    Hint: Add the vector's first component to x and its second component to y.

    Why: Adding 4 to x and subtracting 3 from y maps (−2, 5) to (2, 2).

  2. 2. Reflect the point (−6, 4) in the x-axis.

    Check answer

    Answer: The reflected point is (−6, −4).

    Hint: Use the rule (x, y) to (x, −y).

    Why: Reflection in the x-axis keeps x fixed and changes the sign of y.

  3. 3. Rotate the point (3, −1) through 180 degrees about the origin.

    Check answer

    Answer: The rotated point is (−3, 1).

    Hint: Use the rule (x, y) to (−x, −y).

    Why: A 180-degree origin rotation changes the sign of both coordinates.

  4. 4. Triangle ABC has A(0, 1), B(3, 1) and C(1, 4); translate every vertex by (−2, 3).

    Check answer

    Answer: The image vertices are A′(−2, 4), B′(1, 4) and C′(−1, 7).

    Hint: Apply x minus 2 and y plus 3 separately to A, B and C.

    Why: The same horizontal and vertical changes must be added to every vertex.

  5. 5. Triangle PQR has P(1, 2), Q(5, 2) and R(3, 4); reflect it in the y-axis.

    Check answer

    Answer: The image vertices are P′(−1, 2), Q′(−5, 2) and R′(−3, 4).

    Hint: Change the sign of x for all three points and leave y unchanged.

    Why: Reflection in the y-axis reverses every x-coordinate and preserves every y-coordinate.

  6. 6. A point moves from (−1, 4) to (5, 1) under a translation; determine the translation vector.

    Check answer

    Answer: The translation vector is (6, −3).

    Hint: Subtract each original coordinate from its corresponding image coordinate.

    Why: The coordinate changes are 5 minus −1 equals 6 horizontally and 1 minus 4 equals −3 vertically.

  7. 7. A student rotates only one vertex of a quadrilateral and then redraws the other three by eye; explain the error and give the correct method.

    Check answer

    Answer: The student has not applied one transformation consistently, so every original vertex must be mapped by the stated coordinate rule before the image is joined.

    Hint: Write an image coordinate beside each original vertex before drawing any image edges.

    Why: A transformation of a point set requires the same exact rule at every point.

  8. 8. Rotate triangle A(1, 1), B(4, 1), C(2, 3) by 90 degrees anticlockwise about the origin and state two invariant properties.

    Check answer

    Answer: The image is A′(−1, 1), B′(−1, 4), C′(−3, 2), and its side lengths, angle measures and area are invariant, so any two of these are valid.

    Hint: Map all three coordinates first, then name properties that rigid motion cannot change.

    Why: The rule (x, y) to (−y, x) produces a congruent image under a rigid rotation.

  9. 9. Point P(2, −3) is translated by (4, 1) and then reflected in the x-axis; find the final coordinate and explain why reversing the order gives a different result.

    Check answer

    Answer: Translation gives (6, −2) and reflection gives (6, 2), while reflecting first gives (2, 3) and then translating gives (6, 4), so the order changes the final y-coordinate.

    Hint: Write the intermediate coordinate for each order before comparing the final points.

    Why: A sequence of transformations must be applied in the stated order because the second rule acts on the first image.

  10. 10. Triangle A has vertices (1,1), (4,1), (2,3). Reflect it in the y-axis. Give the image coordinates and state one property preserved by the transformation.

    Check answer

    Answer: The image vertices are (−1,1), (−4,1), (−2,3). Lengths and angles are preserved, while orientation is reversed.

    Hint: Represent the information first, then calculate, interpret and independently check the result.

    Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Practice and reviewReady for practice

Common mistake: Transforming one vertex is enough to transform the entire polygon.

Correction: Every vertex must be mapped by the same rule before the image can be drawn.

Common mistake: The instruction 'rotate 90 degrees' completely determines an image.

Correction: A rotation also requires a centre and direction unless they are already fixed by convention or context.

Common mistake: Reflecting in the y-axis changes the y-coordinate.

Correction: Reflection in the y-axis changes the sign of x while leaving y unchanged.

Curriculum alignmentStart here

We are learning to describe translations, reflections and rotations of point sets using coordinate rules in the Cartesian plane.

A coordinate transformation maps every point of an object to an image according to one consistent rule. A translation adds the same horizontal and vertical changes to every point, a reflection reverses position across a stated mirror line, and a rotation turns every point through the same angle around a stated centre.

Common origin-centred rules include reflection in the y-axis, which maps (x, y) to (−x, y), reflection in the x-axis, which maps (x, y) to (x, −y), a 90-degree anticlockwise rotation, which maps (x, y) to (−y, x), and a 180-degree rotation, which maps (x, y) to (−x, −y). The rule must be applied to every vertex, not just one convenient point.

Translations, reflections and rotations are rigid transformations, so corresponding lengths, angle measures and area remain invariant and the image is congruent to the original. A reflection reverses orientation, while a translation and rotation preserve cyclic orientation; these properties provide checks on calculated coordinates.

Success criteria

  • I can apply one transformation rule to every vertex of a shape.
  • I can specify the vector, mirror line or centre, angle and direction needed to define a transformation.
  • I can compare an object with its image and identify preserved lengths, angles and area.
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