m348-assignment2 - 6. Let O be a dilation with centre O and...

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Math 348 (2006): Assignment #2 due Monday, July 17, 2006 1. Let f be an isometry of the plane. Show that f takes a pair of parallel lines to a pair of parallel lines. (You can use the fact that we know it takes lines to lines, and that it preserves angles.) 2. Consider the isometry r ` which is reflection across a line ` . Which lines are invariant under r ` ? Which circles are invariant under r ` ? Justify your answers. 3. Consider the isometry R O,θ which is counterclockwise rotation by an angle θ about the point O , where 0 < θ < 360 . Are there any lines which are invariant under R O,θ ? Are there any circles which are invariant under R O,θ ? Justify your answers. (It depends on θ .) 4. Consider the translation T ~v , where ~v is a vector (directed line segment). Are there any lines invariant under T ~v ? Are there any circles invariant under T ~v ? Justify your answers. 5. Show that the composition of three half-turns is a single half-turn.
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Unformatted text preview: 6. Let O be a dilation with centre O and scaling factor , where 6 = 0 , 1. Show that O commutes with a reection r ` across a line ` through O . This is called a dilative reection. Show also that O commutes with a rotation R O, about O . This is called a dilative rotation. 7. Let O and O be two dilations about distinct points O and O with scaling factors and , respectively, with 6 = 0 , 1 and 6 = 0 , 1. Suppose further that 6 = 1. Show that the composition O O is again a dilation with scaling factor . Show that the centre P of this dilation lies on the line - OO such that the OP = -1 -1 OO , with P and O on the same side of O if -1 -1 &gt; 0 and P , O on opposite sides of O if -1 -1 &lt; 0. What happens when = 1? 1...
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