AI generatedGeometry and measuresGCSE Higher

Combining transformations

Identify the overall effect and what remains invariant.

  • Match corresponding vertices before describing a transformation.
  • Reflection: mirror line. Rotation: centre, angle and direction. Translation: vector.
  • Enlargement: centre and scale factor. Apply combined transformations in the stated order.

Example 1

Reflect ABC in the -axis, then reflect the image in the -axis. Draw the final image and describe one equivalent transformation.

Coordinate diagram−224−2240ABC
Coordinate diagram
A 180° rotation about the origin.
Black: original. Blue: image.−4−224−4−22460ABCA′B′C′
Black: original. Blue: image.

First (, ) becomes (−, ). The second reflection gives (−, −). Follow the transformations in the stated order.

Example 2

Reflect ABC in the -axis, then reflect the image in the -axis. Draw the final image and describe one equivalent transformation.

Coordinate diagram−2246−22460ABC
Coordinate diagram
A 180° rotation about the origin.
Black: original. Blue: image.−6−4−2246−4−224680ABCA′B′C′
Black: original. Blue: image.

First (, ) becomes (−, ). The second reflection gives (−, −). Follow the transformations in the stated order.

Your turn

Question 1

Reflect ABC in the -axis, then reflect the image in the -axis. Draw the final image and describe one equivalent transformation.

Coordinate diagram−2246−22460ABC
Coordinate diagram
Check answer
A 180° rotation about the origin.
Black: original. Blue: image.−6−4−2246−4−22468100ABCA′B′C′
Black: original. Blue: image.

First (, ) becomes (−, ). The second reflection gives (−, −). Follow the transformations in the stated order.

Question 2

Reflect ABC in the -axis, then reflect the image in the -axis. Draw the final image and describe one equivalent transformation.

Coordinate diagram−224−2240ABC
Coordinate diagram
Check answer
A 180° rotation about the origin.
Black: original. Blue: image.−4−224−4−2240ABCA′B′C′
Black: original. Blue: image.

First (, ) becomes (−, ). The second reflection gives (−, −). Follow the transformations in the stated order.

Question 3

Reflect ABC in the -axis, then reflect the image in the -axis. Draw the final image and describe one equivalent transformation.

Coordinate diagram−2246−22460ABC
Coordinate diagram
Check answer
A 180° rotation about the origin.
Black: original. Blue: image.−6−4−2246−4−22460ABCA′B′C′
Black: original. Blue: image.

First (, ) becomes (−, ). The second reflection gives (−, −). Follow the transformations in the stated order.

Question 4

Reflect ABC in the -axis, then reflect the image in the -axis. Draw the final image and describe one equivalent transformation.

Coordinate diagram−2246−22460ABC
Coordinate diagram
Check answer
A 180° rotation about the origin.
Black: original. Blue: image.−6−4−2246−4−224680ABCA′B′C′
Black: original. Blue: image.

First (, ) becomes (−, ). The second reflection gives (−, −). Follow the transformations in the stated order.

Question 5

Reflect ABC in the -axis, then reflect the image in the -axis. Draw the final image and describe one equivalent transformation.

Coordinate diagram−2246−22460ABC
Coordinate diagram
Check answer
A 180° rotation about the origin.
Black: original. Blue: image.−4−2246−6−4−22460ABCA′B′C′
Black: original. Blue: image.

First (, ) becomes (−, ). The second reflection gives (−, −). Follow the transformations in the stated order.

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