Question

The perimeter of a triangular field is 240 dm. If two of its sides are 78 dm and 50 dm, find the length of the perpendicular on the side of length 50 dm from the opposite vertex.

Answer 1

Given : The perimeter of a triangular field is 240 dm. If two of its sides are 78 dm and 50 dm.

Let the sides be a = 78 dm , b = 50 dm  

Perimeter of ∆ = a + b + c  

⇒ 240 = 78 + 50 + c

⇒ 240 = 128 + c

⇒ c = 240 - 128

⇒ c = 112 dm

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

Semi-perimeter (s) = (78 + 50 + 112)/2

s = 240/2  

s = 120 dm

Using Heron’s formula :  

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

A = √120(120 -  78)(120 - 50)(120 - 112)

A = √120 × (42) × (70) × (8)

A = √(10 × 12) (6 × 7) × (7 × 10) × (2 × 2 × 2)  

A = √(10 × 10)  × (6 × 2) × (6) × (7 × 7) × (2 × 2 × 2)  

A = √(10 × 10)  × (6 × 6) × (7 × 7) × (2 × 2 × 2 × 2 )  

A = 10 × 6 × 7 × 2 × 2

A = 10 × 168

A = 1680 dm²

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

1680 = ½  × 50 × altitude  

Altitude  = (1680 × 2)/50

Altitude  = 1680/25

Altitude  = 67.2 dm  

Hence, the length of the perpendicular on the side of length 50 dm from the opposite vertex is 67.2 dm .

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Answer 2

Answer:

Step-by-step explanation:

Given :-

Perimeter of the triangle = 240 dm

Two sides = 78 dm and 50 dm

Third side  = 240 - (78 + 50) dm = 112 dm

To Find :-

The length of the perpendicular on the side of length 50 dm from the opposite vertex.

Formula to be used :-

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

Area of triangle = ½ x Base x altitude

Solution :-

⇒ s = (a + b + c)/2

⇒ s  = (78 + 50 + 112)/2

⇒ s  = 240/2

⇒ s = 120 dm

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

⇒  Area of the triangle  = √120 (120 - 50)(120 - 78)(120 - 112)

⇒  Area of the triangle  = √120 ×  70 × 42 × 8

⇒   Area of the triangle = √2822400

⇒  Area of the triangle = 1680 dm²

Now, the length of the perpendicular.

⇒ Area of triangle = ½ x Base x altitude

⇒ 1680 = ½  × 50 × altitude  

⇒ Altitude  = (1680 × 2)/50

⇒  Altitude  = 1680/25

⇒  Altitude  = 67.2 dm  

Hence, the length of the perpendicular on the side of length 50 dm from the opposite vertex is 67.2 dm .