hw4sol - Homework 4(not for credit 1 Suppose the lines m n...

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Homework 4 (not for credit) 1. Suppose the lines ‘, m, n have equations X = 2, Y = 3, and Y = 5. Find the equations for σ m σ and σ n σ m . Solution: σ ( x, y ) = (4 - x, y ), σ m ( x, y ) = ( x, 6 - y ) and σ n ( x, y ) = ( x, 10 - y ). It follows that σ m σ ( x, y ) = σ m (4 - x, y ) = (4 - x, 6 - y ) and that σ n σ m ( x, y ) = σ n ( x, 6 - y ) = ( x, 10 - (6 - y )) = ( x, y + 4). 2. Let A = (0 , 0), B = (5 , 0), C = (0 , 10), D = (4 , 2), E = (1 , - 2), F = (12 , - 4). Find lines ‘, ‘ 0 , ‘ 00 so that the products of the reflections in ‘, ‘ 0 , ‘ 00 map 4 ABC to 4 DEF . Solution: This problem was stated incorrectly!!!! you can find two lines and 0 so that the reflection in these two lines sends 4 ABC to 4 DEF (it cannot be done with three). This is the problem we solve below. First we find a reflection σ which maps A to D . The line through (0 , 0) and (4 , 2) has slope 1 2 so must have slope - 2. The line must also contain the midpoint of the line segment from (0 , 0) to (4 , 2) which is (2 , 1). Thus, is the line with equation Y - 1 = - 2( X - 2).
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