PROVING QUADRILATERALS ARE RHOMBUSES

Example :

Show that the quadrilateral KLMN shown below is a rhombus.

Solution :

We can use any of the three ways described in the concept summary above. For instance, we could show that opposite sides have the same slope and that the diagonals are perpendicular. Another way, shown below, is to prove that all four sides have the same length.

Finding the length of LM :

= √{[(2 - (-2)]² + (1 - 3)²}

= √{(2  + 2)² + (- 2)²}

= √{4² + 4}

= √{16 + 4}

= √20

Finding the length of MN :

MN = √{(6 - 2)² + (3 - 1)²}

= √{4² + 2²}

= √{16 + 4}

= √20

Finding the length of NK :

NK = √{(2 - 6)² + (5 - 3)²}

= √{(-4)² + 2²}

= √{16 + 4}

= √20

Finding the length of KL :

KL = √{(-2 - 2)² + (3 - 5)²}

= √{(-4)² + (-2)²}

= √{16 + 4}

= √20

From the above calculations, we have

LM = MN = NK = KL

Hence, KLMN is a rhombus.

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