Question

The perimeter of a triangular field is 450 m and its sides are in the ratio 13 12: 5. Find the area of the triangle.​

Answer 1

Answer :-

• 6750m².

Given :-

• The perimeter of a triangular field is 450m and its sides are in the ratio 13 : 12 : 5

To Find :-

• Area of the triangle

Solution :-

Put x in the ratio

Sides are

13x

12x

5x

As we know that

Perimeter of a triangle is sum of all sides

Now,

13x + 12x + 5x = 450

→ 30x = 450

→ x = 450/30

→ x = 15

Put the value of x in the ratio

• 13x = 13 × 15 = 195
• 12x = 12 × 15 = 180
• 5x = 5 × 15 = 75

• a = 195
• b = 180
• c = 75
• Semi perimeter (S) = 450/2 = 225

As we know that

Area of a triangle is :-

• √ s (s - a) (s - b) (s - c)

According to question :-

√ 225 × (225 - 195) × (225 - 180) × (225 - 75)

→ √ 225 × 30 × 45 × 150

→ √ 45562500

→ 6750m²

Hence, the area of the triangle is 6750m².

Answer 2

Answer :

Given :

• Perimeter of triangular field = 450 m

• Sides of field are in ratio of 13 : 12 : 5

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

To Find :

• Area of triangular field

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

Solution :

• Let the side be x , then the sides become 13x , 12x and 5x .

$\leadsto \sf 13x\:+\:12x\:+\:5x\:=\:450$

$\leadsto \sf 30x\:=\:450$

$\leadsto \sf x\:=\: \dfrac{450}{30}$

$\leadsto \sf x\:=\:15$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• SIDES OF TRIANGLE ARE :

➪ 13x = 13(15) = 195 m

➪ 12x = 12(15) = 180 m

➪ 5x = 5(15) = 75 m

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

★ Now , we have to find area :

• We will be using Heron's formula -

☯︎ Here , a = 195 , b = 180 , c = 75

Now , Semi perimeter = $\sf \dfrac{450}{2} \:=\:225$

$\sf Heron's \:Formula\: =\: \\\small{\sf{\red{ \sqrt{s(s\:-\:a)(s\:-\:b)(s\:-\:c)}}}}$

➪ $\small \sf \sqrt{225(225\:-\:195)(225\:-\:180)(225\:-\:75)}$

➪ $\small \sf \sqrt{225(30)(45)(150)}$

➪ $\small \sf \sqrt{45562500}$

• Area of field = 6750 m²

_____________________