If the diagonals of a rhombus are 32 and 60 cm, find its perimeter.


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Answer:

136 cm

Step by Step Explanation:
  1. We have been given the (length of) diagonals of a rhombus and we are required to find its perimeter. If we can find the side of the rhombus, finding the perimeter will be easy. Let us first find the side of the rhombus.
  2. We know that the diagonals of a rhombus are perpendicular bisectors of each other. Let us use this fact in combination with the Pythagorean Theorem to find the side of the rhombus.
  3. Let us refer to the figure below showing a rhombus ABCD with diagonals of 32 and 60 cm intersecting at point O:
  4. Since the diagonals of a rhombus are perpendicular bisectors of each other, following facts hold true for the triangle COB:
    • Triangle COB is a right angled triangle with CB being the hypotenuse.
    • The length of the side CO = Half of diagonal CA =  
      32
      2
        = 16 cm.
    • The length of the side BO = Half of diagonal BD =  
      60
      2
        = 30 cm.
  5. Using the Pythagorean Theorem for triangle ΔCOB, we can write:
    CB2 = (16)2 + (30)2
    ⇒ CB2 = 256 + 900
    ⇒ CB2 = 1156
    ⇒ CB = √1156
    ⇒ CB = 34
  6. We know that the perimeter of a rhombus is 4 times its side.
    Thus, the perimeter of the rhombus = 4 × CB
    = 4 × 34
    = 136 cm

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