Question

length of adjecent sides of a parallelogram 13 cm and 14 cm . length of a diagonal is 15 cm find the area of the parallelogram.​

Answer 1

Given: Length of two adjacent sides of a parallelogram are 13 cm and 14 cm & Length of a diagonal is 15 cm.

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To find: Area of parallelogram?

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☯ Let ABCD be a parallelogram where area of ∆ ABC and ∆ CAD is equal.

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$\setlength{\unitlength}{1 cm}\begin{picture}(20,15)\thicklines\qbezier(1,1)(1,1)(6,1)\qbezier(1,1)(1,1)(1.6,4)\qbezier(1.6,4)(1.6,4)(6.6,4)\qbezier(6,1)(6,1)(6.6,4)\qbezier(1.6,4)(1.6,4)(6,1)\put(0.7,0.5){\sf B}\put(6,0.5){\sf C}\put(1.4,4.3){\sf A}\put(6.6,4.3){\sf D}\put( - 0.1,2.5){\sf 13 cm}\put(3,0.3){\sf 14 cm}\put(4.1,2.7){\sf 15 cm}\end{picture}$

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Therefore,

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Area or //gm ABCD = Area of ∆ ABC + Area of ∆ CAD

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$\therefore$ Ar. (ABCD) = 2(Ar. ∆ABC)⠀⠀⠀⠀[ Ar.(∆ CAD) = Ar.(∆ ABC) ]

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$\bf{\dag}\;{\underline{\frak{In\: triangle\:ABC,}}}$

$\underline{\bigstar\:\boldsymbol{Using\:Heron's\:Formula\::}}$

$\star\;{\boxed{\sf{\pink{Area_{\;(triangle)} = \sqrt{s(s - a)(s - b)(s - c)}}}}}$

$\sf Where \begin{cases} & \sf{a = AB = \bf{13\:cm}} \\ & \sf{b = BC = \bf{14\:cm}} \\ & \sf{c = AC = \bf{15\;cm}} \end{cases}$

$:\implies\sf s = semi - perimeter$

$:\implies\sf s = \dfrac{a + b + c}{2}$

$:\implies\sf s = \dfrac{13 + 14 + 15}{2}$

$:\implies\sf s = \cancel{\dfrac{42}{2}}$

$:\implies{\underline{\boxed{\frak{\purple{s = 21\:cm}}}}}\;\bigstar$

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$\bf{\dag}\;{\underline{\frak{Now\:Putting\:values\;in\;formula,}}}$

$:\implies\sf Area_{\;(\triangle\:ABC)} = \sqrt{21(21 - 13)(21 - 14)(21 - 15)}$

$:\implies\sf Area_{\;(\triangle\:ABC)} = \sqrt{21 \times 8 \times 7 \times 6}$

$:\implies\sf Area_{\;(\triangle\:ABC)} = \sqrt{7056}$

$:\implies{\underline{\boxed{\frak{\purple{Area_{\;(\triangle\:ABC)} = 84\:cm^2}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Area\:of\: \triangle\:ABC\:is\: \bf{84\:cm^2}.}}}$

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Therefore,

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Area of Parallelogram ABCD,

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$:\implies\sf Area_{\:( ||gm\:ABCD)} = 2 \times Area_{\;(\triangle\:ABC)}$

$:\implies\sf Area_{\:( ||gm\:ABCD)} = 2 \times 84$

$:\implies{\underline{\boxed{\frak{\pink{Area_{\:( ||gm\:ABCD)} = 168\:cm^2}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Area\:of\: parallelogram\:ABCD\:is\: \bf{168\:cm^2}.}}}$

Answer 2

Given:-

• Length of adjacent sides of a parallelogram are 13 and 14.
• Length of the diagonal is 15 cm.

To Find:-

• Find the area of parallelogram

Solution:-

Given sides

️ ➭ a = 13

️ ➭ b = 14

️ ➭ c = 15

So,

️ ➭ Semi perimeter( S ) = (a + b + c)/2

️ ➭ S = (13 + 14 + 15)/2

️ ➭ S = 42/2

$\boxed{\bf{\color{yellw}{ \: S \: = \: 21 \: }}}$

So,

$\tt \implies { \: Area \: = \: \sqrt{s(s \: - \: a)(s \: - \: b)(s \: - \: c} }$

$\tt \implies { \: Area \: = \: \sqrt{21(21 \: - \: 13 \: )(21 \: - \: 14 \: )(21 \: - \: 15 \: } }$

$\tt \implies { \: Area \: = \: \sqrt{ \: 21( \: 8 \: ) \: ( \: 7 \: ) \: ( \: 6 \: )} }$

$\tt \implies { \: Area \: = \: \sqrt{ \: 7056 \: } }$

$\tt \implies { \: Area \: = \: \sqrt{ \: 2( \: 84 ) \: } }$

$\tt \implies { \: Area \: \: of \: parallelogram \: = \: 168 \: }$