Question

$\rm\underline\bold{ QUESTION \red{\huge{\checkmark}}}$the perimete tryr of a triangle is 450m . the ratio of i ints side's are 13:12:5 using herons formula find the area of the triangle​

Answer 1

$\huge\mathtt{\fbox{\red{Answer✍︎}}}$

$\large\underline\mathfrak{\pink{GIVEN,}}$

$\dashrightarrow \red{ Perimeter\:of\: triangle= 450m}$

$\dashrightarrow \orange{ratios\:of\:the\:sides\:of\:triangles\:are }\\ \dashrightarrow \pink{13:12:5}$

$\large{\boxed{\bf{ \mathfrak{\blue{FORMULA,}}}}}$

$\rm{\boxed{\sf{ \large{\circ}\:\: area\:of\:triangle_{(herons\:fomula)}= \sqrt{s(s - a)(s - b)(s - c)} \:\: \large{\circ}}}}$

$\large\underline\mathfrak{\pink{TO\:FIND,}}$

$\dashrightarrow \green{Area\:of\:triangle\:using\:herons\: formula.}$

$\large\underline\mathfrak{\purple{SOLUTION,}}$

$\therefore \green{let\:the\:constant\:be\:'x'\:m}$

$\dashrightarrow \red{sides\:of\:triangle\: are,}\\ \dashrightarrow \blue{a=13x} \\ \purple{b= 12x }\\ \green{c= 5x }$

$\therefore \orange{finding\:the\:value\:of\:x.}$

$\dashrightarrow \purple{perimeter\:of\:triangle= a+b+c}$

$\dashrightarrow \blue{13x+12x+5x=450}$

$\implies \green{30x=450}$

$\implies \green{x= \dfrac{450}{30} }$

$\implies \green{x = \cancel\dfrac{450}{30}}$

$\implies \green{x=15}$

$\rm{\boxed{\bf{ \:\: x=15 \:\: }}}$

$\dashrightarrow \red{x=15}\\ \pink{a= 13x= 13\times 15= 195m}\\ \blue{\dashrightarrow b=12x= 12\times 15 = 180m}\\ \dashrightarrow \purple{c=5x= 5\times 15= 75m}$

NOW, FINDING AREA OF TRIANGLE BY HERONS FORMULA,

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

$\implies \orange{ s= \dfrac{195+180+75}{2}}$

$\implies \orange{s= \dfrac{450}{2}}$

$\implies \orange{s= 225}$

$\bf\dashrightarrow \red{area\:of\:triangle_{(herons\:fomula)}= \sqrt{s(s - a)(s - b)(s - c)}}$

$\implies \purple{\sqrt{225(225-195)(225-180)(225-75)}}$

$\implies \purple{ \sqrt{225(30)(45)(150)}}$

$\implies \purple{\sqrt{ 225(1350)(150)}}$

$\implies \purple{15\sqrt{202500}}$

$\implies \purple{15 \times 450}$

$\implies \purple{6750m^2}$

$\rm{\boxed{\sf{ \large{\circ}\:\:area\:of\:triangle= 6750m^2 \:\: \large{\circ}}}}$

$\rm\underline\mathfrak{\pink{AREA\:OF\:TRIANGLE\:IS\:6750m^2.}}$

Answer 2

Explanation:

\rm{\boxed{\sf{ \large{\circ}\:\: area\:of\:triangle_{(herons\:fomula)}= \sqrt{s(s - a)(s - b)(s - c)} \:\: \large{\circ}}}}

∘areaoftriangle

(heronsfomula)

=

s(s−a)(s−b)(s−c)

∘

\large\underline\mathfrak{\pink{TO\:FIND,}}

TOFIND,

\dashrightarrow \green{Area\:of\:triangle\:using\:herons\: formula.}⇢Areaoftriangleusingheronsformula.

\large\underline\mathfrak{\purple{SOLUTION,}}

SOLUTION,

\therefore \green{let\:the\:constant\:be\:'x'\:m}∴lettheconstantbe

′

x

′

m

\begin{gathered}\dashrightarrow \red{sides\:of\:triangle\: are,}\\ \dashrightarrow \blue{a=13x} \\ \purple{b= 12x }\\ \green{c= 5x } \end{gathered}

⇢sidesoftriangleare,

⇢a=13x

b=12x

c=5x

\therefore \orange{finding\:the\:value\:of\:x.}∴findingthevalueofx.

\dashrightarrow \purple{perimeter\:of\:triangle= a+b+c}⇢perimeteroftriangle=a+b+c

\dashrightarrow \blue{13x+12x+5x=450}⇢13x+12x+5x=450

\implies \green{30x=450}⟹30x=450

\implies \green{x= \dfrac{450}{30} }⟹x=

30

450

\implies \green{x = \cancel\dfrac{450}{30}}⟹x=

30

450

\implies \green{x=15}⟹x=15

\rm{\boxed{\bf{ \:\: x=15 \:\: }}}

x=15

\begin{gathered}\dashrightarrow \red{x=15}\\ \pink{a= 13x= 13\times 15= 195m}\\ \blue{\dashrightarrow b=12x= 12\times 15 = 180m}\\ \dashrightarrow \purple{c=5x= 5\times 15= 75m} \end{gathered}

⇢x=15

a=13x=13×15=195m

⇢b=12x=12×15=180m

⇢c=5x=5×15=75m

NOW, FINDING AREA OF TRIANGLE BY HERONS FORMULA,

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

2

a+b+c

\implies \orange{ s= \dfrac{195+180+75}{2}}⟹s=

2

195+180+75

\implies \orange{s= \dfrac{450}{2}}⟹s=

2

450

\implies \orange{s= 225}⟹s=225

\bf\dashrightarrow \red{area\:of\:triangle_{(herons\:fomula)}= \sqrt{s(s - a)(s - b)(s - c)}}⇢areaoftriangle

(heronsfomula)

=

s(s−a)(s−b)(s−c)

\implies \purple{\sqrt{225(225-195)(225-180)(225-75)}}⟹

225(225−195)(225−180)(225−75)

\implies \purple{ \sqrt{225(30)(45)(150)}}⟹

225(30)(45)(150)

\implies \purple{\sqrt{ 225(1350)(150)}}⟹

225(1350)(150)

\implies \purple{15\sqrt{202500}}⟹15

202500

\implies \purple{15 \times 450}⟹15×450

\implies \purple{6750m^2}⟹6750m

2

\rm{\boxed{\sf{ \large{\circ}\:\:area\:of\:triangle= 6750m^2 \:\: \large{\circ}}}}

∘areaoftriangle=6750m

2

∘