Question

The lengths of the sides of a triangle are in the ratio 3 : 4 : 5 and its perimeter is 144 cm. Find the area of the triangle and the height corresponding to the longest side.

Answer 1

Given : The lengths of the sides of a triangle are in the ratio 3 : 4 : 5 and its perimeter is 144 cm.

Let the sides be a = 3x , b = 4x and c = 5x .

Perimeter of ∆ = a + b + c  

⇒ 144 = 3x + 4x + 5x

⇒ 12x = 144

⇒ x = 144/12

⇒ x = 12

So , the Sides of a triangle are :  

a = 3x = 3 × 12 = 36 m

b = 4x = 4 × 12 = 48 m

c = 5x = 5 × 12 = 60 m

Semi Perimeter of the ∆,s = (a + b + c) /2

Semi-perimeter (s) = (36 + 48 + 60)/2

s = 144/2  

s = 72 m

Using Heron’s formula :  

Area of the ∆ , A = √s (s - a) (s - b) (s - c)

A = √72(72 -  36)(72 - 48)(72 - 60)

A = √72 × (36) × (24) × (12)

A = √(36 × 2) (36) (12 × 2) × 12

A = √(36 × 36 × 12 × 12) × (2 × 2)

A = 36 × 12 × 2

A = 72 × 12

A = 864 cm²

Now, area of triangle , A = ½ x Base x altitude

864 = ½ × 60 × altitude

[longest side = 60 cm]

864 = 30 × altitude

Altitude = 864 / 30

Altitude = 28.8 cm

The height corresponding to the longest side is 28.8 cm.

Hence, the area of the triangle is 864 cm² and the height corresponding to the longest side is 28.8 cm.

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Answer 2

Answer:

Step-by-step explanation:

Given :-

Ratio of sides = 3 : 4 : 5

Perimeter of the triangle = 144 cm.

To Find :-

Area of the triangle and the height corresponding to the longest side.

Formula to be used :-

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

Solution :-

Let the sides of triangles are 3x, 4x and 5x.

⇒ 3x + 4x + 5x = 144

⇒ 12x = 144

⇒ x = 12

Then, Sides are triangle are 3(12) = 36 cm, 4(12) = 48 cm , 5(12) = 60 cm.

Now, Area of triangle,

s =72, a = 36, b = 48, c = 60

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

Area of triangle =  √72(72 - 36)(72 - 48)(72 - 60)

Area of triangle = √72 (36) (24) (12)

Area of triangle = √746496

Area of triangle = 864 cm²

Here, the longest side is 60 cm.

Area of triangle = ½ x Base × altitude

864 = ½ × 60 × altitude

864 = 30 × altitude

Altitude = 864/30

Altitude = 28.8 cm

Hence,  the area of the triangle is 864 cm²  and the height corresponding to the longest side is 28.8 cm.

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