Question

prove diagonals in rhombus are perpendicular bisectors of each other,using vectors

Answer 1

\[(A+B)\cdot(B-A)=A\cdot B-A\cdot A+B\cdot B-B\cdot A\\\quad =A\cdot B -|A|^2+|B|^2-A\cdot B=|B|^2-|A|^2 = 0\]

Answer 2

Therefore the diagonals are penpendicular

Answer 3

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Answer 4

\[\cos x=\frac{B\cdot(A+B)}{|B||A+B|}=\frac{A\cdot B + |B|^2}{|B||A+B|}\\\cos y=\frac{A\cdot(A+B)}{|A||A+B|}=\frac{A\cdot B + |A|^2}{|B||A+B|}\]
because \[|A| = |B|\] and \[|A|^2=|B|^2\] the two fractions are equal. therefore, cos x = cos y, or that the two angles x and y are congruent.