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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.

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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