Question

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​ by​

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

$\large{\underline{ \underline{ \sf{ \maltese \: {Given:-}}}}}$

• The perimeter of a triangle = 450 m.
• The ratio of sides of triangle = 13:12:5

⠀⠀⠀⠀⠀⠀⠀

$\large{\underline{ \underline{ \sf{ \maltese \: {To \: find:-}}}}}$

• Area of triangle using Heron's formula.

$\large{\underline{ \underline{ \sf{ \maltese \: {Solution:-}}}}}$

⠀⠀⠀⠀⠀⠀⠀

Let:–

• Sides of triangle = 13x, 12x and 5x.

Where:–⠀⠀⠀⠀⠀⠀⠀

$\qquad\bull \sf \: {a = \bf{13x}}$

$\qquad\bull \sf \: {b= \bf{12x}}$

$\qquad\bull \sf \: {c= \bf{5x}}$

$\qquad \quad{:}\longrightarrow\sf 13x + 12x + 5x = 450$

$\qquad \quad{:}\longrightarrow\sf 30x = 450$

$\qquad \quad{:}\longrightarrow\sf x = \cancel\dfrac{450}{30}$

$\qquad \quad { : }\longrightarrow \underline{\boxed{\sf x = 15}}$

Therefore:–

$\qquad\bull \sf \: {a = 13x = \underline{\underline{195 m}}}$

$\qquad\bull \sf \: {b = 12x = \underline{\underline{180m}}}$

$\qquad\bull \sf \: {c = 5x = \underline{\underline{75m}}}$

Using Heron's Formula:–

We know that,

$\qquad \quad{:}\longrightarrow\sf {s = \dfrac{a + b + c}{2} }$

$\qquad \quad{:}\longrightarrow\sf {s = \dfrac{195 + 180 + 75}{2} }$

$\qquad \quad{:}\longrightarrow\sf {s = \dfrac{450}{2} }$

$\qquad \quad{:}\longrightarrow \underline{ \boxed{\sf {s = 225 }}}$

Now,

⠀⠀⠀⠀⠀⠀⠀

$\large \boxed{\boxed{\sf{Area_{\;(triangle)} = \sqrt{s(s - a)(s - b)(s - c)}}}}$

$\qquad \quad{:}\longrightarrow\sf \sqrt{225 \bigg(225 - 195 \bigg) \bigg (225 - 180 \bigg) \bigg(225 - 75 \bigg)}$

$\qquad \quad{:}\longrightarrow\sf \sqrt{225 \bigg(30 \bigg) \bigg (45\bigg) \bigg(150 \bigg)}$

$\qquad \quad{:}\longrightarrow\sf \sqrt{225 \times202500 }$

$\qquad\quad{:}\longrightarrow\sf \sqrt{ 45562500}$

$\qquad\quad{:}\longrightarrow \underline{\boxed{\sf { 6750 \: {m}^{2}}}}$

$\large{\underline{ \underline{ \sf{ \maltese \:{Answer:-}}}}}$

• The area of the triangle = 6750 square metres