Preston says the sequence of transformations that maps triangle ABC onto triangle DEF is: rotate it 180 degrees about the origin, and then translate it up 7 units. Chanel says the sequence of transformations that maps triangle ABC onto triangle DEF is: reflect it across the x-axis, then reflect it across the y-axis, and then translate it up 6 units and left 1 unit.

Part A
First, take a look at Preston’s sequence. How does a 180-degree rotation about the origin change the coordinates of a shape?

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Part B
Complete the table to find the rule for the rotation, the coordinates of triangle ABC, and the coordinates of the rotated image, triangle A′B′C′.

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Part C
How does a translation 7 units up change the coordinates of a shape?

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Part D
Complete the table to find the rule for the translation, the coordinates of triangle A′B′C′, and the coordinates of the translated image, triangle A″B″C″.

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Part E
Now look at Chanel’s sequence. How does a reflection across the x-axis change the coordinates of a shape?

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Part F
Complete the table to find the rule for the reflection, the coordinates of triangle ABC, and the coordinates of the reflected image, triangle A′B′C′.

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Part G
How does a reflection across the y-axis change the coordinates of a shape?

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Part H
Complete the table to find the rule for the reflection, the coordinates of triangle A′B′C′, and the coordinates of the reflected image, triangle A″B″C″.

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Part I
How does a translations 6 units up and 1 unit left change the coordinates of a shape?

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Part J
Complete the table to find the rule for the translation, the coordinates of triangle A″B″C″, and the coordinates of the translated image, triangle A‴B‴C‴.

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Part K
Whose sequence proves triangle ABC is congruent to triangle DEF? Why?