Question

18. The sides of a triangle are in the ratio of 3 : 5 : 7 and its perimeter is 300 cm. Its area will be:

1000√3 sq.cm
1500√3 sq.cm
1700√3 sq.cm
1900√3 sq.cm
​

Answer 1

Given sides of a ∆ are in ratio 3:5:7. Perimeter of ∆ is 300 cm

We have to find area of ∆.

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☯ Let sides of ∆ be 3x, 5x and 7x.

⠀⠀⠀⠀

⋆ Reference of image is shown in diagram

$\setlength{\unitlength}{0.8cm}\begin{picture}(6,5)\thicklines\put(1,0.5){\line(2,1){3}}\put(4,2){\line(-2,1){2}}\put(2,3){\line(-2,-5){1}}\put(0.7,0.3){ \sf A }\put(4.05,1.9){ \sf B }\put(1.7,2.95){ \sf C }\put(3.1,2.5){ \sf 3x }\put(1,1.7){ \sf 5x }\put(2.9,1.05){ \sf 7x }\end{picture}$

⠀⠀⠀⠀

$\dag\;{\underline{\frak{We\;know\;that,}}}$

$\star\;\bf{\boxed{\sf{\pink{Perimeter_{\;(triangle)} = a + b + c}}}}$

where,

a, b and c are the sides of Triangle.

⠀⠀⠀⠀

$\dag\;{\underline{\frak{Putting\;values\;:}}}$

$:\implies\sf 3x + 5x + 7x = 300$

$:\implies\sf 15x = 300$

$:\implies\sf x = \cancel{ \dfrac{300}{15}}$

$:\implies{\underline{\boxed{\sf{\pink{x = 20}}}}}\;\bigstar$

Therefore,

Sides of ∆ are,

• 3x = 3 × 20 = 60 cm

• 5x = 5 × 20 = 100 cm

• 7x = 7 × 20 = 140 cm

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☯ Now, Finding area of triangle,

⠀⠀⠀⠀

$\star\;\bf{\boxed{\sf{\purple{area_{\;(triangle)} = \sqrt{s(s - a)(s - b)(s - c)}}}}}$

where,

s = semi - perimeter

$\sf \star\; s = \dfrac{a + b + c}{2}$

$:\implies\sf s = \dfrac{60 + 100 + 140}{2}$

$:\implies\sf s = \dfrac{300}{2}$

$:\implies\bf s = 150\;cm$

$\dag\;{\underline{\frak{Putting\;values\;in\; Formula\;:}}}$

$:\implies\sf \sqrt{150(150 - 60)(150 - 100)(150 - 140)}$

$:\implies\sf \sqrt{150(90)(50)(10)}$

$:\implies\sf \sqrt{150 \times 90 \times 50 \times 10}$

$:\implies\sf \sqrt{6750000}$

$:\implies{\underline{\boxed{\sf{\purple{1500 \sqrt{3}\;cm^2}}}}}\;\bigstar$

$\therefore\;{\underline{\sf{Area\;of\;\triangle\;is\; \bf{1500 \sqrt{3}\;cm^2}}}}$

Answer 2

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

• The sides of a triangle are in the ratio of 3:5:7 .
• The perimeter is 300 can.

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

• Area of the triangle .

$\underline{\underline{\sf{\maltese\:Formula\:used\::-}}}$

• Semi-perimeter = a + b + c/2
• Area of triangle = √s(s - a) (s - b) (s - c)

$\underline{\underline{\sf{\maltese\:Concept\::-}}}$

$\odot$ Here we have given that the sides of the triangle are in the ratio of 3:5:7. as we know that to find the area we need sides of the triangle so firstly we will find out the sides of the triangle.

$\odot$ After finding the sides we will find out the semi-perimeter of the triangle by using the formula ( Semi-perimeter = a + b + c/2 )

$\odot$ Now after finding the semi-perimeter of the triangle. we will find out the area of the triangle by substituting the given values in the formula Area of triangle = √s(s - a) (s - b) (s - c)

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

$\bigstar$ Let us find out the sides of the triangle.

$\underline{\underline{\sf{\odot\:Assumption\:Needed\::-}}}$

Let the sides of the triangle be

• 3x
• 5x
• 7x

$\qquad\sf{:\implies\:3x\:+\:5x\:+\:7x\:=\:300}$

$\qquad\sf{:\implies\:15x\:=\:300}$

$\qquad\sf{:\implies\:x\:=\:\dfrac{300}{15}}$

$\qquad\sf{:\implies\:x\:=\:20}$

$\underline{\underline{\sf{\maltese\:Sides\::-}}}$

$\qquad\sf{:\implies\:Side\:(a)\:=3x}$

$\qquad\sf{:\implies\:Side\:(a)\:=3\:\times\:20}$

$\qquad\sf{:\implies\:Side\:(a)\:=\:60\:cm}$

$\qquad\sf{:\implies\:Side\:(b)\:=\:5x}$

$\qquad\sf{:\implies\:Side\:(b)\:=\:5\:\times\:20}$

$\qquad\sf{:\implies\:Side\:(b)\:=\:100\:cm}$

$\qquad\sf{:\implies\:Side\:(c)\:=\:7x}$

$\qquad\sf{:\implies\:Side\:(c)\:=\:7\:\times\:20}$

$\qquad\sf{:\implies\:Side\:(c)\:=\:140\:cm}$

Hence the sides of the triangle are 60cm,100cm and 140cm

$\bigstar$ Let us find out the semi-perimeter of the triangle by substituting the sides in the formula ( Semi-perimeter = a + b + c/2 )

$\qquad\sf{:\implies\:Semi\:-\:perimeter\:=\:\dfrac{a\:+\:b\:+\:c}{2}}$

$\qquad\sf{:\implies\:Semi\:-\:perimeter\:=\:\dfrac{60\:+\:100\:+\:140}{2}}$

$\qquad\sf{:\implies\:Semi\:-\:perimeter\:=\:\dfrac{300}{2}}$

$\qquad\sf{:\implies\:Semi\:-\:perimeter\:=\:150\:cm}$

Hence the semi-perimeter of the triangle is 150cm.

$\bigstar$ Let us find out the Area of the triangle by substituting the values in the formula.

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

• s = semi=perimeter
• a = length of the triangle a
• b = length of the triangle b
• c = length of the triangle c

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

$\qquad\sf{:\implies\:Area\:=\:\sqrt{150\:(\:150\:-\:60\:)\:(\:150\:-\:100\:)\:(\:150\:-\:140\:)}}$

$\qquad\sf{:\implies\:Area\:=\:\sqrt{150\:(\:90\:)\:(\:50\:)\:(\:10\:)}}$

$\qquad\sf{:\implies\:Area\:=\:\sqrt{5\:\times\:3\:\times\:10\:\times\:3\:\times\:3\:\times\:10\:\times5\:\times\:10\:\times\:10}}$

$\qquad\sf{:\implies\:Area\:=\:100\:\times\:5\:\times\:3\sqrt{3} }$

$\qquad\sf{:\implies\:Area\:=\:1500\:\sqrt{3}\:cm^{2} }$