Question

The sides of a triangular plot are in the ratio 4:5:6 and it's perimeter is 150 cm. Find it's area​

Answer 1

Answer:

$\sf{The \ area \ of \ the \ triangular \ plot \ is}$

$\sf{375\sqrt7 \ cm^{2}}$

Given:

$\sf{\leadsto{The \ sides \ of \ triangular \ plot \ are \ in \ the}}$

$\sf{ratio \ of \ 4:5:6}$

$\sf{\leadsto{The \ perimeter \ of \ the \ plot \ is \ 150 \ m}}$

To find:

$\sf{The \ area \ of \ the \ triangular \ plot.}$

Solution:

$\sf{Let \ the \ constant \ be \ n}$

$\sf{According \ to \ the \ first \ condition. }$

$\sf{Sides \ are \ 4n, \ 5n \ and \ 6n}$

$\sf{According \ to \ the \ second \ condition. }$

$\sf{4n+5n+6n=150}$

$\sf{\therefore{15n=150}}$

$\sf{\therefore{n=\dfrac{150}{15}}}$

$\sf{\therefore{n=10}}$

$\sf{The \ sides \ are}$

$\sf{\mapsto{4n=4(10)=40 \ cm,}}$

$\sf{\mapsto{5n=5(10)=50 \ cm,}}$

$\sf{\mapsto{6n=6(10)=60 \ cm}}$

$\sf{s=\dfrac{Perimeter}{2}}$

$\sf{\therefore{s=\dfrac{150}{2}}}$

$\sf{\therefore{s=75}}$

$\sf{By \ heron's \ formula}$

$\sf{A(\triangle)=\sqrt{s(s-40)(s-50)(s-60)}}$

$\sf{\therefore{A(\triangle)=\sqrt{75(75-40)(75-50)(75-60)}}}$

$\sf{\therefore{A(\triangle)=\sqrt{75\times35\times25\times15}}}$

$\sf{\therefore{A(\triangle)=\sqrt{75\times75\times5\times5\times7}}}$

$\sf{\therefore{A(\triangle)=375\sqrt7 \ cm^{2}}}$

$\sf\purple{\tt{\therefore{The \ area \ of \ the \ triangular \ plot \ is}}}$

$\sf\purple{\tt{375\sqrt7 \ cm^{2}}}$

Answer 2

EXPLANATION.

Sides of a triangular plot are In the ratio

=> 4:5:6

=> it's perimeter = 150 cm.

To find it's area.

Let the sides of a triangular plot =

=> 4x , 5x , 6x

=> 4x + 5x + 6x = 150

=> 15x = 150

=> x = 10

=> 4x = 4 X 10 = 40

=> 5x = 5 X 10 = 50

=> 6x = 6 X 10 = 60

=> Area of semi perimeter = a + b + c / 2

=> 40 + 50 + 60 / 2

=> 150/2 = 75 = s

$\rm \to \green{\: area \: of \: triangle \: = \sqrt{s(s - a)(s - b)(s - c)} }$

$\rm \to \: \sqrt{75(75 - 40)(75 - 50)(75 - 60)}$

$\rm \to \: \sqrt{75(35)(25)(15)}$

$\rm \to \: \sqrt{984375}$

$\rm \to \: 992.156 \: cm {}^{2}$