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**Composition of Rigid Motions**

A sequence of basic rigid motions (translation, rotation, and reflection) “Teaching Geometry According to the Common Core Standards”, H. Wu, 2012

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**We start with two identical figures.**

What sequence of basic rigid motions maps the left H exactly onto the right H so that all corresponding angles and segments coincide?

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**For clarity, we’ll dash the lines of the original H’s**

and show in red the image of the left H that will be the subject of our rigid motions.

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We draw a straight line segment (a vector) from the lower right corner of the left H to the upper left corner of the right H .

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A translation along this vector maps the lower right corner of the red H onto the upper left corner of the right H.

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A clockwise rotation of 90 degrees around the red dot maps the right side of the red H to the top side of the right H.

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**Then a reflection across the top of the right H**

maps the red H exactly onto the right H.

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To recap: the composition of a translation along a vector, a rotation around a point,

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To recap: the composition of a translation along a vector, a rotation around a point, and a reflection across a line is a sequence of basic rigid motions that maps the left H exactly onto the right.

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Angle A B C side angle A0A0 C0C0 B0B0 side angle Angle-Side-Angle Congruence by basic rigid motions A geometric realization of a proof in H. Wu’s “Teaching.

Angle A B C side angle A0A0 C0C0 B0B0 side angle Angle-Side-Angle Congruence by basic rigid motions A geometric realization of a proof in H. Wu’s “Teaching.

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