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Let ABCD be a quadrilateral such that AB + AD + CB + CD = 0 and | AC | = | BD | . Then ABCD is necessarily a

Options

  1. ARhombus
  2. BParallelogram
  3. CSquare
  4. DRectangle

Correct answer

D. Rectangle

Step-by-step solution

Let the position vectors of the vertices A, B, C, and D be a , b , c , and d respectively. The given vector equation is: AB + AD + CB + CD = 0 Expressing these in terms of position vectors: ( b - a ) + ( d - a ) + ( b - c ) + ( d - c ) = 0 Simplifying the equation: 2 b + 2 d - 2 a - 2 c = 0 2( b + d ) = 2( a + c ) a + c = b + d Dividing both sides by 2 gives: a + c 2 = b + d 2 This shows that the position vector of the midpoint of diagonal AC is equal to the position vector of the midpoint of diagonal BD . Since th

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