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The diagonals of rhombus measure 16cm and 30cm. Find its perimeter.

Answer Verified Verified
Hint – Here first we draw a rhombus after that to determine its perimeter we will use the concept that Diagonals of rhombus bisect perpendicular to each other and Perimeter of rhombus \[\;4 \times side\].

Complete step-by-step answer:

AC and BD are diagonals of a rhombus.
AC=16, BD=30
We know that diagonals of rhombus are perpendicular bisectors of each other.
$\therefore $ AO = 8 and BO = 15
In right angled \[\Delta AOB\]
\[\begin{array}{*{20}{l}}
  { \Rightarrow \;\;{{\left( {AB} \right)}^2} = {{\left( {AO} \right)}^2} + {{\left( {BO} \right)}^2}} \\
  { \Rightarrow \;\;{{\left( {AB} \right)}^2} = {{\left( 8 \right)}^2} + {{\left( {15} \right)}^2}} \\
  { \Rightarrow \;\;{{\left( {AB} \right)}^2} = 64 + 225} \\
  { \Rightarrow \;\;{{\left( {AB} \right)}^2} = 289} \\
  { \Rightarrow \;\;AB = 17cm}
\end{array}\]
We know that all four sides of rhombus are equal.
$\therefore $ Perimeter of rhombus ABCD \[ = {\text{ }}4 \times AB\]
 \[\begin{array}{*{20}{l}}
  {\; = {\text{ }}4 \times 17} \\
  { = {\text{ }}68cm}
\end{array}\]


Note – In this problem, first we calculate the side of the rhombus with help of Pythagoras theorem then calculate the perimeter of the rhombus by using the concept which is given above.
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