Question

The perimeter of a triangular field is 270 cm and its sides are in the ratio 25 : 17 : 12. Find its area.

Answer 1

Answer:

Step-by-step explanation:

Given :-

Perimeter = 270 cm

Ratio of sides = 25 : 17 : 12

To Find :-

Area of triangle

Formula to be used :-

Heron's formula i.e,  √s(s - a)(s - b)(s - c)

Solution :-

Let the sides be 25x , 17x and 12x.

⇒ 25x + 17x + 12x = 270

⇒ 54x = 270

⇒ x = 270/54

⇒ x = 5

1st side = 25x = 25 × 5 = 125 cm

2nd side =  17x = 17 × 5 = 85 cm

3rd side = 12x = 12 × 5 = 60 cm

Semi-perimeter, s = 270/2 = 135 cm

Now, area of triangle

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

⇒ Area of triangle = √135(135 - 125)(135 - 85)(135 - 60)

⇒ Area of triangle = √135 (10) (50) (75)

⇒ Area of triangle = √5 × 9 × 3 × 5 × 2 × 25 × 2 × 3 × 25

⇒ Area of triangle = 2250 cm²

Hence, the Area of triangle is 2250 cm².

Answer 2

$\Large{\underline{\underline{\mathfrak{\tt{\red{Question}}}}}}$

The perimeter of a triangular field is 270 cm and its sides are in the ratio 25 : 17 : 12. Find its area.

$\Large{\underline{\underline{\mathfrak{\tt{\red{Solution}}}}}}$

$\Large{\underline{\mathfrak{\tt{\orange{Given}}}}}$

• The perimeter of a triangular field is 270 cm
• its sides are in the ratio 25 : 17 : 12

$\Large{\underline{\mathfrak{\tt{\orange{Find}}}}}$

• Area of triangle

$\Large{\underline{\underline{\mathfrak{\tt{\orange{Explanation}}}}}}$

Using Formula

$\boxed{\small{\tt{\green{\:Perimeter\:of\:triangle\:=\:Sum\:of\:all\:side}}}}$

Let, Here

• First Side be = 25x cm
• Second Side be = 17 cm
• Third Side be = 12 cm

Then,

==> Perimeter of triangle = (25+17+12) * x

==> 270 = 54x

==> x = 270/54

==> x = 5

$\Large{\underline{\mathfrak{\tt{\orange{Hence}}}}}$

• First Side be = 25x = 25 * 5 = 125 cm
• Second Side be = 17x = 17 * 5 = 85 cm
• Third Side be = 12x = 12 * 5 = 60 cm

___________________________

Now, Calculate Area of Triangle

Assume that , Here ∆ ABC

where,

• AB = 125 cm
• BC = 85 cm
• CA = 60 cm

By, Heron's Formula

$\boxed{\small{\tt{\green{\:semi\:perimeter(S)\:=\:\dfrac{AB+BC+CA}{2}}}}}$

==> S = (125+85+60)/2

==>S = 270/2

==> S = 135 cm

Now, Area will be

$\boxed{\small{\tt{\green{\:Area_{triangle}\:=\:\sqrt{S(S-AB)(S-BC)(S-CA)}}}}} \\ \\ \\ \mapsto\tt{\:Area_{triangle}\:=\:\sqrt{135(135-125)(135-85)(135-60)}} \\ \\ \mapsto\tt{\:Area_{triangle}\:=\:\sqrt{135*10*50*75}} \\ \\ \mapsto\tt{\:Area_{triangle}\:=\:\sqrt{5*9*3*5*2*25*2*3*25}} \\ \\ \mapsto\tt{\:Area_{triangle}\:=\:(5*5*5*3*3*2)} \\ \\ \mapsto\bold{\tt{\red{\:Area_{triangle}\:=\:2250\:cm^2\:\:\:\:\:Ans}}}$

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