Question

Q.12 The sides of a triangle are in the ratio of 3:5:7 and its perimeter is 300 cm. Its area will be: a) 1000✓3. b)1700✓3. c)1900✓3. d)1500✓3​

Answer 1

Answer :

• Area of triangle is 1500√3 cm²

Given :

• The sides of a triangle are in the ratio of 3:5:7
• Perimeter is 300 cm

To find :

• Area

Solution :

Given,the sides of a triangle are in the ratio of 3:5:7 so,

• Let the sides of triangle be 3x , 5x and 7x

Perimeter is 300cm so,

➟ 3x + 5x + 7x = 300

➟ 15x = 300

➟x = 300/15

➟ x = 20cm

Sides are :

• 3x = 3(20) = 60 cm
• 5x = 5(20) = 100 cm
• 7x = 7(20) = 140 cm

Finding the semi Perimeter of triangle :

We know that

• Semi perimeter of triangle = a + b + c / 2

Where,

• a is 60 cm
• b is 100 cm
• c is 140 cm

Substituting the value :

➟ s = a + b + c / 2

➟ s = 60 + 100 + 140 / 2

➟ s = 300/2

➟ s = 150cm

Semi perimeter of triangle is 150cm

Finding the Area of triangle :

We know that

• Area of triangle = √s(s - a) (s - b) (s - c)

Where,

• s is semi perimeter of triangle
• a , b , c is a sides of triangle

Substituting the value :

➟Area of triangle = √s(s - a) (s - b) (s - c)

➟ √150(150 - 60) (150 - 100) (150 - 140)

➟ √150(90) (50) (10)

➟ √5 × 3 × 10 × 3 × 3 × 10 × 5 × 10 × 10

➟ 100 × 5 × 3√3

➟ 1500√3 cm²

Hence, Area of triangle is 1500√3 cm²

Answer 2

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

• The sides of a triangle are in the ratio of 3:5:7 .

• The perimeter is 300 cm.

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

• 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

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

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

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

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

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

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

• Hence the area of the triangle is 1500√3 cm²

$\underline{\underline{\sf{\maltese\:Extra\:Information\::-}}}$

• Area of triangle = √s(s - a) (s - b) (s - c) is known as herons formula .

where

• s is semi-perimeter

• a is the length of the triangle a
• b is the length of the triangle b
• c is the length of the triangle c