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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?

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.

The Matrix

Investigate the transfomations of the plane given by the 2 by 2 matrices with entries taking all combinations of values 0. -1 and +1.

Rots and Refs

Age 16 to 18
Challenge Level

The point $P$ has coordinates $(r\cos \phi, r\sin \phi)$ where the distance of $P$ from the origin is $OP=r$ and the line $OP$ is at angle $\phi$ to the x axis. Find the image of this point under the transformation given by the matrix

$\mathbf{T_1}=\left( \begin{array}{cc} cos \theta & -\sin \theta\\ sin \theta & cos \theta \end{array} \right)$

Draw a diagram and describe the effect of this transformation on the points of the plane.

The point $P$ has coordinates $(p,q)$ and the point $P'$ is the reflection of $P$ in the line $y = x\tan\theta$. In the diagram below the lines $P'X'$ and $XA'$ are perpendicular to the line $OA'X'$. "


Prove that $OX = OX' = p$, $P'X' = PX = q$ and $OA = OA' = p\cos 2\theta.$ Find the lengths $BP',\ AX'$ and $BX'$ and hence prove that transformation given by the matrix

$\mathbf{T_2}=\left( \begin{array}{cc} cos 2\theta & sin2\theta\\ sin2\theta & -\cos2\theta \end{array} \right)$

gives a reflection in the line $y=\tan\theta$.