Question

Find the area of a triangle , two sides of which are 7cm and 10cm and the perimeter is 30cm.​

Answer 1

$\large\underline\bold{\mathcal{\orange{ANSWER}} }$

$\rm{\boxed{\sf{ Option:- (a) \: 10\sqrt{3} \:is\:correct }}}$

$\rm\underline{\underline{\bold{\pink{\mathfrak{explanation\:in\:detail:-}}}}}$

$\large\underline\bold{GIVEN:-}$

$\dashrightarrow side(a)= 7cm \\ \dashrightarrow side(b)=10cm \\ \dashrightarrow perimeter\:of\:triangle\: = 30cm$

$\large\underline\bold{\pink{TO\:FIND:-}}$

$\dashrightarrow Area\:of\:triangle\: \pink{ {}^{\circ}} \purple{\smile} \pink{ {}^{\circ}}$

$\large\underline{\underline\bold{ANSWER:}}$

$\therefore \: we \:dont\: know \: third \:side\:for\: \\ finding\:the\:area\:of\:rectangle \\ \ therefore \: finding\:the\:third \:side \\ \dashrightarrow We\:have\: perimeter\:and\:two\:sides\:of\:triangle \\ \therefore \rm{\boxed{\underline{ perimeter=a+b+c }}} \\ \therefore by \: using\: the \: formula \\ : \therefore let\:the\:third\:side\: be \: x\:cm \\ \ therefore 7+10+x= 30 \\ :\implies 17+x=30 \\ :\implies x=30-17 \\ :\implies x= 13 \\ \boxed{\red{ x=13cm}}$

$\therefore Now \:finding\: area\:of\:triangle \: \\ we \: have , \\ \\ \dashrightarrow a= 7xm \\ \dashrightarrow b= 10cm \\ \dashrightarrow c= 13cm$

$\\ \\ \rm{\boxed{\underline{ \purple{ \mathcal{ area\:of\:triangle_{\pink{(herons\:fomula)}}= \sqrt{s(s-a)(s-b)(s-c)}}}}}}$

$\\ \\ \therefore finding\: s\: \\ \therefore s= \dfrac{a+b+c}{2} \:\:\: or \: \dfrac{perimeter}{2} \\ :\dashrightarrow \: s= \dfrac{30}{2} \\ :\dashrightarrow \: s=15cm \\ \bf{\boxed{ \red{ S=15}}}$

$\\ \\ \therefore finding:-\:\rm{\mathcal{\red{\underline{area\:of\: triangle}}}} \\ :\implies \sqrt{15(15-7)(15-10)(15-13) }$

$: \implies \sqrt{ 15(8)(5)(2)} \\ :\implies \sqrt{ 30 \times 40 } \\ :\implies \sqrt{1200} \\ :\implies \sqrt{ \underline{ 2\times 2\times } \: \underline{ 2\times 2\times } \:\: \underline{5\times 5}\:\: \times 3} \\ :\implies 2\times 5\sqrt{ 3} \\ :\implies 10\sqrt{ 3} cm^2 \\ \\ \rm{\boxed{ \underline{ \red{ 10\sqrt{3} cm^2}}}}$

$\rm{\underline{\bold{ \underline{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:}}}}$