z_1, z_2, z_3, z_4 are distinct complex numbers representing the vertices of a quadrilateral ABCD taken in order. If z_1 - z_4 = z_2 - z_3 and arg[ dfrac((z_4 - z_1))((z_2 - z_1))] = pi/2, then the quadrilateral is A rectangle B rhombus C square D trapezium

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z_1, z_2, z_3, z_4 are distinct complex numbers representing the vertices of a quadrilateral ABCD taken in order. If z_1 - z_4 = z_2 - z_3 and arg[ dfrac((z_4 - z_1))((z_2 - z_1))] = pi/2, then the quadrilateral is A rectangle B rhombus C square D trapezium

Answer 1

Let $A=z_{1}$ $B=z_{2}$ $C=z_{3}$ $D=z_{4}$ Hence $z_{1}-z_{4}=z_{2}-z_{3}$ Implies $AD||BC$ Hence opposites sides are parallel. Therefore its a type of parallelogram. Now $arg(\dfrac{z_{4}-z_{1}}{z_{2}-z_{1}})=\angle A$ $=\dfrac{\pi}{2}$. Hence All the angles are $90^{0}$ therefore, it is rectangular a rectangle

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z_1, z_2, z_3, z_4 are distinct complex numbers representing the vertices of a quadrilateral ABCD taken in order. If z_1 - z_4 = z_2 - z_3 and arg[ dfrac((z_4 - z_1))((z_2 - z_1))] = pi/2, then the quadrilateral is A rectangle B rhombus C square D trapezium

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