Question

the length of the sides of a triangle are in the ratio of 4:5:6 and its perimeter is 150cm. Find its area ​

Answer 1

$\large\underline\bold{ANSWER \huge{\checkmark}}$

$\large\underline\bold{GIVEN,}$

$\dashrightarrow sides\:of\:triangle\:are\:given\:in\:ratio\\ \dashrightarrow 4:5:6$

$\dashrightarrow taking \:side\:in\:ratio\:as\:x$

$\dashrightarrow a= 4x$

$\dashrightarrow b=5x$

$\dashrightarrow c= 6x$

✯FORMULA IN USE,

$\large{\boxed{\bf{ \star\:\:area\:of\:triangle_{Herons\:formula}= \sqrt{s(s-a)(s-b)(s-c)} \:\: \star}}}$

$\dashrightarrow perimeter\:of\:triangle= 150cm$

$\large\underline\bold{TO\:FIND,}$

$\dashrightarrow area\:of\:triangle$

$\large\underline\bold{SOLUTION,}$

$\dashrightarrow Perimeter\:of\:triangle= a+b+c$

$\implies 150=4x+5x+6x$

$\implies 150= 15x$

$\implies x= \dfrac{150}{15}$

$\implies x= \cancel\dfrac{150}{15}$

$\implies x= 10cm$

$\dashrightarrow a=4x=4\times 10= 40cm\\ \dashrightarrow b=5x=5\times 10=50cm \\ \dashrightarrow c= 6x= 6\times 10= 60cm$

NOW , FINDING AREA OF TRIANGLE,

$\therefore s= \dfrac{a+b+c}{2}$

$\dashrightarrow s= \dfrac{40+50+60}{2}$

$\dashrightarrow s=\dfrac{150}{2}$

$\dashrightarrow s=\cancel\dfrac{150}{2}$

$\dashrightarrow s=75$

$\large{\boxed{\bf{ \star\:\:s=75 \:\: \star}}}$

$\therefore Area\:of\:triangle= \sqrt{s(s-a)(s-b)(s-c)}$

$\implies \sqrt{75(75-40)(75-50)(75-60)}$

$\implies \sqrt{75(35)(25)(15)}$

$\implies 5\sqrt{75(35)(15)}$

$\implies 5\sqrt{75(525)}$

$\implies 5\sqrt{39375}$

$\implies 5\times 198.43$

$\implies 992.15cm^2$

$\large{\boxed{\bf{ \star\:\: 992.15cm^2\:\: \star}}}$

$\large\underline\bold{AREA\:OF\:TRIANGLE\:IS\:992.15cm^2}$

____________

Answer 2

Step-by-step explanation:

Answer: Let the sides of the triangle be 4x,5x,6x, respectively. Therefore, the sides are 40 cm, 50 cm, and 60 cm.