Question

Sanya has a piece of land which is in the shape of a rhombus. She wants her one daughter and one son to work on the
land and produce different crops to suffice the needs of their family. She divided the land in two equal parts. If the
perimeter of the land is 400 m and one of the diagonals is 160 m, then the area each of them will get​

Answer 1

⠀⠀⠀⠀☯ Let ABCD be the field which is divided by the diagonal BD = 160 m into two equal parts.

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$\setlength{\unitlength}{1.2 cm}\begin{picture}(0,0)\thicklines\qbezier(0,0)(0,0)(1,3)\qbezier(3,0)(3,0)(4,3)\qbezier(1,3)(1,3)(4,3)\qbezier(3,0)(0,0)(0,0)\put(-0.4,-0.2){\sf D}\put(3.2,-0.2){\sf C}\put(4.2,3.1){\sf B}\put(0.6,3.1){\sf A}\qbezier(0,0)(2,1.5)(4,3)\put(2.2,3.2){\sf 100 m}\put(1.2, - 0.4){\sf 100 m}\put( - 0.5,1.5){\sf 100 m}\put(3.8,1.4){\sf 100 m}\put(1.2,1.6){\sf 160 m}\end{picture}$

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Since ABCD is a rhombus of perimeter 400 m.

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Therefore,

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$:\implies\sf AB = BC = CD = DA = \dfrac{400}{4}\;m = \bf{100\;m}$

☯ Let s be the semi - perimeter of ∆ BCD.

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$:\implies\sf s = \dfrac{BC + CD + BD}{2}$

$:\implies\sf s = \dfrac{100 + 100 + 160}{2}$

$:\implies\sf s = \dfrac{360}{2}$

$:\implies{\boxed{\frak{\pink{s = 180\;m}}}}\;\bigstar$

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☯ Now, Finding Area of ∆BCD,

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Using Heron's Formula,

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$\star\;{\boxed{\sf{\purple{A = \sqrt{s(s - a)(s - b)(s + c)}}}}}$

$:\implies\sf A = \sqrt{180(180 - 100)(180 - 100)(180 - 160)}$

$:\implies\sf A = \sqrt{180 \times 80 \times 80 \times 20}$

$:\implies{\boxed{\frak{\pink{A = 4800\;m^2}}}}\;\bigstar$

$\therefore\;{\underline{\sf{Hence,\;each\;of\;the\;two\; children\;will\;get\;an\;area\;of\; \bf{4800\;m^2}.}}}$