Question

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Answer 1

» Question :

Sides of a triangle are in the ratio of 12 : 17 : 25 and it's Perimeter is 540 cm . Find it's area.

» To Find :

The Area of the triangle :

» We Know :

Heron's formula :

$\sf{\underline{\boxed{A_{t} = \sqrt{s(s - a)(s - b)(s - c)}}}}$

Where,

• s = Semi-perimeter, i.e,

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

• a = side of the triangle
• b = side of the triangle
• c = side of the triangle

Perimeter of a triangle :

$\sf{\underline{\boxed{P_{t} = a + b + c}}}$

Where ,

• a = side of the triangle
• b = side of the triangle
• c = side of the triangle

» Concept :

To Find the area of the triangle , first we have to find the the sides of the triangle .

ATQ

Let the side be x .

By using the formula , and putting the value in it ,we get :

$\sf{\underline{\boxed{P_{t} = a + b + c}}}$

$\sf{\Rightarrow 540 = 12x + 17x + 25x}$

$\sf{\Rightarrow 540 = 12x + 17x + 25x}$

$\sf{\Rightarrow 540 = 54x}$

$\sf{\Rightarrow \dfrac{540}{54} = x}$

$\sf{\Rightarrow \dfrac{\cancel{540}}{\cancel{54}} = x}$

$\sf{\Rightarrow 10 = x}$

Hence ,the value of x is 10 .

Now Putting the value of x in the given sides, i.e

Sides of the triangle :

• $a = 12x \rightarrow 12 \times 10 \rightarrow 120 cm$

• $b = 17x \rightarrow 17 \times 10 \rightarrow 170 cm$

.

• $c = 25x \rightarrow 25 \times 10 \rightarrow 250 cm$

Hence ,the sides of the triangle are ,120 cm ,170 cm and 250 cm.

Now ,by this information we can find the area of the triangle.

» Solution :

• a = 120 cm
• b = 170 cm
• c = 250 cm

Semi-perimeter :

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

Putting the value and by solving it ,we get :

$\sf{\Rightarrow s = \dfrac{120 + 170 + 250}{2}}$

$\sf{\Rightarrow s = \dfrac{540}{2}}$

$\sf{\Rightarrow s = \dfrac{\cancel{540}}{\cancel{2}}}$

$\sf{\Rightarrow s = 270 cm}$

Area of the triangle :

Formula :

$\sf{\underline{\boxed{A_{t} = \sqrt{s(s - a)(s - b)(s - c)}}}}$

Now , Substituting the value and solving it ,we get :

$\sf{\Rightarrow A_{t} = \sqrt{270(270 - 120)(270 - 170)(270 - 250)}}$

$\sf{\Rightarrow A_{t} = \sqrt{270 \times 150 \times 100 \times 20}}$

$\sf{\Rightarrow A_{t} = \sqrt{81000000}}$

$\sf{\Rightarrow A_{t} = 9000 m^{2}}$

Hence ,the area of the triangle is 9000m².

» Additional information :

• Area of a right-angled triangle = ½ab

• Area of an equilateral triangle = $\dfrac{\sqrt{3}a^{2}}{4}$

• Area of a rectangle = legth × breadth

• Area of a square = (side)²

Answer 2

⭐ SOLUTION :-

Let a common be x.

So, The Sides Will be :-

12x, 17x, 25x.

According To The Question,

Perimeter = Sum Of A Triangle = 540cm.

So,

12x + 17x + 25x = 540.

☞ 54x = 540.

☞ x = 540/54.

☞ x = 10.

Hence, Value Of X = 10.

The Sides Of Triangle Are :-

• 12x = 120
• 17x = 170
• 25x = 250.

Now, Calculating Area.

We will Use Heron's Formula.

First, Calculate Semi Perimeter.

Semi-Perimeter = a+b+c/2

Here,

a = 120,

b = 170,

c = 250.

☞ 120+170+250/2 = 270.

Heron's Formula :- √s(s-a)(s-b)(s-c)

Put The Values.

☞ √270(270-120)(270-170)(270-250).

☞ √270 × 150 × 100 × 20.

☞ 9000m²

Hence, Area = 9000m²