a) Show that if two plane figures A and B are congruent, then they are also proj

Question

a) Show that if two plane figures A and B are congruent, then they are also projectively equivilant. (Hint: Use the definition of congruence)

Projectively Equivalent: The geometrical configurations A and B are projectively equivalent if we can arrive from A to B through a succesion of perspectives.

Congruent: Two configurations are congruent if and only if we can find a finite number of succesive reflections to transform one figure into the other.

Congruence in the plane: a transformation brought about by a succession of reflections in lines.

b) Show that two similar triangles are projectively equivilant.

Explanation / Answer

Note that ?(x^2/2 + y^2/2) = ; so the Fundamental Theorem for Line Integrals applies. For both (a) and (b): t = 0 ==> (x, y) = (0, 0), and t = 8 ==> (x, y) = (8, 64). Therefore, ?c F · dX = (x^2/2 + y^2/2) {for (x, y) = (0, 0) to (8, 64)} = 2080.

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