Critical Thinking The sum of two distinct complexnumbers, and can be foundgeometrically

Chapter 10, Problem 49

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Critical Thinking The sum of two distinct complexnumbers, and can be foundgeometrically by means of the so-calledparallelogram rule: Plot the points andin the complex plane and form theparallelogram, three of whose vertices are 0,and as in the figure below. Then the fourthvertex of the parallelogram is the point whosecoordinate is the sumComplete the following proof of the parallelogramrule when anda. Find the slope of the line K from 0 toHint: K contains the points (0, 0) and (a, b).b. Find the slope of the line N from 0 toc. Find the equation of the line L, throughand parallel to line N of part b. Hint: The point(a, b) is on L; find the slope of L by using part band facts about the slope of parallel lines.d. Find the equation of the line M, throughand parallel to line K of part a.e. Label the lines K, L, M, and N in each figure.f. Show by using substitution that the pointsatisfies both the equation of lineL and the equation of line M. Therefore,lies on both L and M. Becausethe only point on both L and M is the fourthvertex of the parallelogram, this vertex mustbe Hence, this vertex hascoordinate