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

✯ Given ✯

• Perimeter of triangle = 450 m
• Ratio of its side = 13 : 12 : 5

✯ To find ✯

• Area of triangle by using Heron's Formula

꧁ Solution ꧂

Concept of the question:

• Here the question is given from Heron's formula. In this question we have given that a triangle has 450 perimeter and also ratio of sides are given 13 : 12 : 5. We have asked to find area of triangle.

Procedure to do question.

• First Here we have to take all sides of triangle's side Ratio in x form as 13x, 12x, 5x, also it will be added to give value of x and then by putting value of x in all sides which in unknown that we give length of all sides. It will be divided by 2 to give semiperimeter of triangle. This semiperimeter will be used in Heron's formula to give area of triangle.

Formula we use :-

${\large{\underline{\boxed{\bf{ Semiperimeter = \frac{ a + b + c}{2}}}}}}$

Here a, b, c are sides.

${\large{\underline{\boxed{\bf{ Area = \sqrt{s(s - a)(s - b)(s - c)} }}}}}$

Here s is semiperimeter and a, b, c are all sides.

● Value of x is :-

${\sf{\implies{13x + 12x + 5x = 450}}}$

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

$\\ {\sf{\implies{x = \frac{450}{30} }}}$

$\large{ \boxed{\implies{ \blue { \bf{x = 15}}}}}$

● So, length of all sides are :-

${\bf{\large{\implies 13x = 13\times 15 = {\large{\boxed{\blue{ \bf{195 \: m}}}}}}}}$

${\bf{\large{\implies 12x = 12\times 15 = {\large{\boxed{\blue{ \bf{180 \: m}}}}}}}}$

${\bf{\large{\implies 5x = 5\times 15 = {\large{\boxed{\blue{ \bf{75 \: m}}}}}}}}$

● Semiperimeter of triangle is :-

• Here a is of 195m , b is of 180m and c is of 75m.

$\\ { \sf{ Semiperimeter = \frac{ a + b + c}{2}}}$

$\\ { \sf{ Semiperimeter = \frac{ 195 + 180 + 75}{2}}}$

$\\ { \sf{ Semiperimeter = \frac{ 450}{2}}}$

$\large{ \boxed{{ \blue { \bf{Semiperimeter = 225 \: m}}}}}$

● Area of triangle is :-

• Here s is semiperimeter which is 225 m.

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

${\sf{ Area = \sqrt{225 \: (225 - 195)(225 - 180)(225 - 75)} }}$

${\sf{ Area = \sqrt{225 \: (30)(45)(150)} }}$

${\sf{ Area = \sqrt{45 \times 5 \times 30 \times 45 \times 30 \times 5} }}$

${\sf{ Area =45 \times 5 \times 30 }}$

$\huge{ \boxed{{ \red { \bf{Area = 6750 \: m^{2}}}}}}$