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# Square Pair

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You might like to look at Flipping Twisty Matrices before investigating this problem.

A 2D constant matrix $M$ acts on the plane, transforming each point with position vector ${\bf r}$ to another point with position vector $M{\bf r}$.

A particular square $S$ has corners with coordinates $(0,0), (0,1), (1,1), (1,0)$.

The action of the matrix $M$ on the points making up the square $S$ produces another shape in the plane.

What quadrilaterals can I transform $S$ into?

What quadrilaterals can I not transform $S$ into?

What shapes other than quadrilaterals can I transform $S$ into?

Another square $T$ has coordinates $(1,2), (2, 1), (3, 2), (2, 3)$.

When will $S$ and $T$ transform into the same type of quadrilateral? When will they transform into two different types of quadrilateral? Construct the matrices $M$ in each case.

*You might like to use this Matrix Transformation tool to test out your ideas.*

*There are more matrix problems in this feature.*

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?

Follow hints using a little coordinate geometry, plane geometry and trig to see how matrices are used to work on transformations of the plane.

Follow hints to investigate the matrix which gives a reflection of the plane in the line y=tanx. Show that the combination of two reflections in intersecting lines is a rotation.