Math, asked by riyabhagoria, 1 month ago

find the area of the rhombus whose perimeter is 80m and One of whose diagonal is 24m​

Answers

Answered by Anonymous
58

\huge\tt{\bold{\purple{\underbrace{\red{\mathfrak{ɾҽզմíɾҽժ⠀ αղsաҽɾ♡༄}}}}}}

\huge{\fbox{\pink{Given:-}}}

Perimeter of the Rhombus = 80m

Diagonal of the Rhombus = 24 m

\huge{\fbox{\pink{To find:-}}}

Area of the Rhombus.

\huge{\fbox{\pink{Solution:-}}}

Perimeter = 80 m,

So, length of a side = 80/4

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀=20m,

and Diagonal of a rhombus (d1) = 24m,

As, we know that,

The halves of diagonals and a side of a rhombus from a right angled triangle with side as hypotenuse.

Let the length of the other diagonal (d2) =2x.

 {x}^{2}  + ( \frac{22}{2} ) ^{2}   =  {20}^{2}  \\   \implies {x}^{2}  + 144 \:  = 400 \\  \implies  {x}^{2}  = 400 - 144 \\   \implies {x}^{2}  = 256 \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \\  \implies \: x =  \sqrt{256}  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \\  \implies \: x = 16 \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:

The measure of the another diagonal of the rhombus (d2) = 2 x 16

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ = 32 m

area \: of \: the \: rhombus  =  \frac{1}{2}  \times d1 \times d2 \\  \implies area \: of \: the \: rhombus =  \frac{1}{2}  \times 32 \times 24 \\  \implies \:area \: of \: the \: rhombus = 384 {m}^{2}  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:

\huge{\fbox{\pink{Formula Used:-}}}

area \: of \: the \: rhombus  =  \frac{1}{2}  \times d1 \times d2

Answered by Anonymous
77

\huge{\underline{\underline{\mathfrak{\red{Given\::}}}}}

  • \bf{Perimeter\:of\:rhombus\:=\:80\:m}
  • \bf{One\:diagonal\:of\:rhombus\:=\:24\:m}

{ }

\huge{\underline{\underline{\mathfrak{\red{Find\::}}}}}

  • \bf{Area\:of\:rhombus}

{ }

\huge{\underline{\underline{\mathfrak{\red{Solution\::}}}}}

  • \bf{\bold{Perimeter\: of\: rhombus \:= \:4 \:\times\: side}}

\:\:\:\:\:\:\:\:\:\:\:\:\::\:\Longrightarrow{\sf{4a\:=\:80}}

\:\:\:\:\:\:\:\:\:\:\:\:\::\:\Longrightarrow{\sf{a\:=\:20\:m}}

\:\:\:\:\:\sf{Let\:AC\:=\:25\:m}

\:\:\:\:\:\sf{\therefore\:OA\:=\:{\frac{1}{2}}}

\:\:\:\:\:\sf{AC\:=\:{\frac{1}{2}}\:\times\:24\:=\:12\:m}

\:\:\:\:\:\bf{\bold{In\:\triangle\:AOB}}

\:\:\:\:\:\bf{OB}^{2}\:{=\:AB}^{2}\:{-\:OA}^{2}

  • \bf{\bold{[By\:using\:Pythagoras\:therorem]}}

\:\:\:\:\:\::\:\Longrightarrow{\sf{OB\:=\:\sqrt{20}^{2}\:{-\:12}^{2}}}

\:\:\:\:\:\:\:\:\:\:\:\:\::\:\Longrightarrow{\bf{\sqrt{400\:-\:144}}}

\:\:\:\:\:\:\:\:\:\:\:\:\::\:\Longrightarrow{\bf{\sqrt{256}}\:=\:16\:m}

  • \:\:\:\:\:\bf{Also\:BO\:=\:OD}

\:\:\:\:\:\bf{[Diagonal\:of\:rhombus\:bisect\:each\:other\:at\:90}^{°}]

\:\:\:\:\:\:\::\:\Longrightarrow{\bf{\therefore\:BD\:=\:20\:B\:=\:2\:\times\:16\:=\:32\:m}}

  • \sf{\therefore\:Area\:of\:rhombus\:=\:{\frac{1}{2}}\:\times\:BD\:\times\:AC}

\:\:\:\:\:\:\::\:\Longrightarrow\sf{\therefore\:Area\:of\:rhombus\:=\:{\frac{1}{2}}\:\times\:32\:\times\:24\:=\:384\:m}^{2}

{ }

\:\:\:\:\therefore{\underline{\sf{Hence,\:the\:area\:of\:rhombus\:is\:{\bold{384\:m}^{2}}}}}

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