Question

A triangle has sides 35 cm, 54 cm and 61 cm long. Find its area. Also, find the smallest of its altitudes.

Answer 1

Given : A triangle has sides 35 cm, 54 cm and 61 cm long.

Let the given sides be a = 35 cm, b = 54  cm , c = 61 cm

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

s = (35 + 54 + 61) / 2

s = 150/2

s = 75 cm

Semi Perimeter of the ∆ = 75 cm

Using Heron’s formula :  

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

A = √75(75 - 35) (75 - 54) (75 - 61)

A = √75 × 40 × 21 × 14

A = √(25 × 3) × (10 × 4) × (3 × 7) × (2 × 7)

A = √25 × 4 × (3 × 3) × (7 × 7) × (2 × 5) × 2

A = √25 × 4 × (3 × 3) × (7 × 7) × (2 × 2) × 5

A = 5 × 2 × 3 × 7 × 2 √5

A = 420 √5

A = 420 × 2.236

A = 939.15 cm²

Area of triangle = 939.15 cm²

Altitude on side 54 cm :  

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

939.15  = ½  × 54 × altitude  

Altitude  = (939.15 × 2)/54

Altitude  = 34.78 cm

Altitude on side 35 cm :  

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

939.15  = ½  × 35 × altitude  

Altitude  = (939.15 × 2)/35

Altitude  = 53.66 cm

 

Altitude on side 61 cm :  

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

939.15  = ½  × 61 × altitude  

Altitude  = (939.15 × 2)/61

Altitude  = 30.79 cm

Smallest altitude = 30.79 cm

Hence, the area of the triangle is 939.15 cm² and the smallest altitude is 30.79 cm.

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Answer 2

Answer:

Step-by-step explanation:

Given :-

The sides of a triangle are, a = 35 cm, b = 54 cm and c = 61 cm

To Find :-

Its area and smallest of its altitudes.

Formula to be used :-

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

Area of triangle = ½ x Base x altitude

Solution :-

Semi Perimeter of the triangle = (a + b + c)/2

⇒ s = (35 + 54 + 61) / 2

⇒  s = 150/2

⇒  s = 75 cm

Putting all the values, we get

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

⇒ Area of the triangle = √75(75 - 35)(75 - 54)(75 - 61)

⇒ Area of the triangle = √75 × 40 × 21 × 14

⇒ Area of the triangle = 939.14 cm²

Now, the smallest of its altitude.

Here, longest side is 61 cm.

⇒ Area of triangle = ½ x Base x altitude

⇒ 939.15  = ½  × 61 × altitude  

⇒ Altitude  = (939.15 × 2)/61

⇒  Altitude  = 30.79 cm

Hence, The area of triangle is 939.14 cm² and the smallest of its altitude is 30.79 cm.