The amazing quadrilateral propertycoordinate free The

Chapter 11, Problem 83

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The amazing quadrilateral propertycoordinate free The points P, Q, R, and S, joined by the vectors u, v, w, and x, are the vertices of a quadrilateral in _3. The four points neednt lie in a plane (see figure). Use the following steps to prove that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram. The proof does not use a coordinate system. S x v u P Q m n w R a. Use vector addition to show that u + v = w + x. b. Let m be the vector that joins the midpoints of PQ and QR. Show that m = 1u + v2>2. c. Let n be the vector that joins the midpoints of PS and SR. Show that n = 1x + w2>2. d. Combine parts (a), (b), and (c) to conclude that m = n. e. Explain why part (d) implies that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram. 8