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8 Methods for Three by One
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?
Rots and Refs
Follow hints using a little coordinate geometry, plane geometry and trig to see how matrices are used to work on transformations of the plane.
Reflect Again
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.
Age 16 to 18
Challenge Level
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 $(0,0), (0,1), (1,0), (1,1)$.
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 $(4,4), (6, 2), (8, 4), (6, 6)$.
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.
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NRICH is part of the family of activities in the Millennium Mathematics Project.
NRICH is part of the family of activities in the Millennium Mathematics Project.