Question

The side of a triangle are in the ratio 5:12:13 and its perimeter is 150. find the area of the triangle

Answer 1

$\huge\it\red{\underline{\underline{Answer:}}}$

$\sf{The \ area \ of \ the \ triangle \ is \ 750 \ unit^{2}}$

$\huge\it\orange{Given:}$

$\sf{\leadsto{The \ sides \ of \ a \ triangle \ are \ in}}$

$\sf{the \ ratio \ 5:12:13 \ and \ it's \ perimeter \ is \ 150}$

$\sf{units.}$

$\huge\it\pink{To \ find:}$

$\sf{Area \ of \ the \ triangle.}$

$\huge\it\green{\underline{\underline{Solution:}}}$

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

$\sf{\therefore{Sides \ of \ triangle \ are:}}$

$\sf{\leadsto{5n, \ 12n \ and \ 13 \ n \ respectively.}}$

$\boxed{\sf{Perimeter=Sum \ of \ all \ sides}}$

$\sf{\therefore{5n+12n+13n=150}}$

$\sf{\therefore{30n=150}}$

$\sf{\therefore{n=\dfrac{150}{30}}}$

$\sf{\therefore{n=5}}$

$\sf{The \ sides \ are:}$

$\sf{\leadsto{5n=5(5)=25 \ units,}}$

$\sf{\leadsto{12n=12(5)=60 \ units,}}$

$\sf{\leadsto{13n=13(5)=65 \ units.}}$

$\sf{Now,}$

$\sf{a=25, \ b=60 \ and \ c=65}$

$\sf{s=semi \ perimeter=\dfrac{150}{2}=75}$

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

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

$\sf{=\sqrt{75(75-25)(75-60)(75-65)}}$

$\sf{=\sqrt{75(50)(15)(10)}}$

$\sf{=\sqrt{25\times25\times30\times30}}$

$\sf{=25\times30}$

$\sf{\therefore{Area \ of \ triangle=750 \ unit^{2}}}$

$\sf\purple{\tt{\therefore{The \ area \ of \ the \ triangle \ is \ 750 \ unit^{2}}}}$

Answer 2

Explanation:

Given:

1. The side of a triangle are in the ratio 5:12:13

2. The perimeter of a triangle is 150 units

To find:

The area of the triangle

Formula:

Perimeter = a+b+c

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

$s=\frac{a+b+c}{2}$

S is called the semi-perimeter of the triangle

Solution:

==> The sides of the triangle are 5:12:13

==> The sides are a,b and c

==> a = 5x

==> b = 12x

==> c = 13x

==> Perimeter = sum of three sides

==> Perimter = a+b+c

==> Perimeter = 150 units

==> Perimeter = 5x+12x+13x

==> 150 = 30x

==> x = 150÷30

==> x=5

==> a= 5x

==> b = 12x

==> c = 13x

==> a = 25

==> b = 60

==> c = 65

==> Apply a,b and c values to find s

==> s is called the semi-perimeter of the triangle

==> $s=\frac{a+b+c}{2}$

==> $s=\frac{25+60+65}{2}$

==> $s=\frac{150}{2}$

==> s = 75

==> $Area =\sqrt{s(s-a)(s-b)(s-c)}$

==>$Area =\sqrt{75(75-25)(75-60)(75-65)}$

==>$Area =\sqrt{75(50)(15)(10)}$

==>$Area =\sqrt{75(750)(10)}$

==> $Area =\sqrt{750\times750}$

==> Area = 750 units²

==> The Area of the triangle is 750 units²