Question

The area of a triangle is 150 cm2and its sides are in the ratio 3 : 4 : 5. What is its perimeter?

Answer 1

Given : -

Area of a triangle = 150 cm²

The sides of the triangle are in the ratio of 3 : 4 : 5

Required to find : -

• Perimeter of the traingle ?

Formula used : -

$\boxed{\rm{ Area \ of \ the \ triangle = \sqrt{ s ( s - a ) ( s - b ) ( s - c ) } }}$

Here,

s = semi - perimeter

a , b , c = 3 sides of the traingle

Solution : -

Area of a triangle = 150 cm²

The sides of the triangle are in the ratio of 3 : 4 : 5

we need to find the perimeter of the triangle .

Since,

we know that ;

Area of the triangle = ½ x base x height

But,

If the measurement of the height is not given then we should we use the Herons formula to find the area of the triangle .

So,

Herons formula

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

However,

The sides of the triangle are in the ratio of 3 : 4 : 5

So,

Let the sides of the triangle be 3x , 4x & 5x

Now,

Using the Herons formula let's find the value of x .

Let's find the semi - perimeter .

s = 3x + 4x +5x/2

s = 12x/2

s = 6x

• Semi - perimeter ( s ) = 6x

According to problem ;

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

This implies ;

=> 150 = √6x ( 6x - 3x )( 6x - 4x )( 6x - 5x )

=> 150 = √6x ( 3x ) ( 2x ) ( x )

=> 150 = √36x⁴

=> 150 = √( 6x² )²

=> 150 = 6x²

=> 6x² = 150

=> x² = 150/6

=> x² = 25

=> x = √25

=> x = ±5

Since,

Length can't be negative

=> x = 5 cm

Hence,

• value of x = 5 cm

Now,

Let's find the measurements of the sides ;

====> 3x = 3 ( 5 ) = 15 cm

===> 4x = 4 ( 5 ) = 20 cm

==> 5x = 5 ( 5 ) = 25 cm

Hence,

The sides of the triangle are 15cm , 20 cm & 25 cm

Now,

Let's find the perimeter of the triangle .

we know that ;

Perimeter of the triangle = sum of all its sides

=> Perimeter = 15 + 20 + 25

=> Perimeter = 60 cm

Therefore ,

Perimeter of the triangle = 60 cm

Answer 2

$\mathcal{\huge{\fbox{\red{Question\:?}}}}$

✴ The area of a triangle is 150 cm2and its sides are in the ratio 3 : 4 : 5. What is its perimeter?

$\mathcal{\huge{\fbox{\green{Answer:-}}}}$

➡ The Perimeter of a triangle is 60 cm.

$\mathcal{\huge{\fbox{\purple{Solution:-}}}}$

Given :-

• The area of a triangle is 150 cm2.

• Its sides are in the ratio 3 : 4 : 5.

To find :-

• Perimeter of the traingle .

Calculation : -

• The Area of a triangle = 150 cm²

• The sides of the triangle are in the ratio of 3 : 4 : 5

Using herons Formulae :-

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

According to the question, The sides of the triangle are in the ratio of 3 : 4 : 5.

Let the sides be x,

So, the sides of the triangle be 3x , 4x & 5x.

Now, finding the semi - perimeter of Δ.

s = a+b+c /2

s = 3x + 4x +5x/2

s = 12x/2

s = 6x

Hence, Semi - perimeter ( s ) = 6x

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

• Area of triangle = 150cm²
• s=6x
• a=3x,b=4x&c=5x

Solving ,

=> 150 = √6x ( 6x - 3x )( 6x - 4x )( 6x - 5x )

=> 150 = √6x ( 3x ) ( 2x ) ( x )

=> 150 = √36x⁴

=> 150 = √( 6x² )²

=> 150 = 6x²

=> 6x² = 150

=> x² = 150/6

=> x² = 25

=> x = √25

=> x = ±5

So, the sides of triangle are

• 3x = 3×5 = 15cm

• 4x = 4×5 = 20cm

• 5x = 5×5 =25cm

Perimeter of a triangle = (a + b + c)

= (15 + 20 + 25)

= (35+25)

= 60cm.

So, The Perimeter of a triangle is 60cm.

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