Question

The Perimeter of a triangle is 50 m. one side of triangle is 4 m longer than the smaller side and third side is 6m less than the twice the smaller side find area of triangle​

Answer 1

Solution

Given :-

• The Perimeter of a triangle is 50 m
• one side of triangle is 4 m longer than the smaller side
• third side is 6m less than the twice the smaller side

Find :-

• area of triangle

Explanation

Let,

• Smaller Side(Second Side) of triangle = x

So, A/C to question,

(one side of triangle is 4 m longer than the smaller side)

==> First Side = (x+4)

Again,

(third side is 6m less than the twice the smaller side )

==> Third Side = 2x-6

Using Formula

★ Perimeter of triangle = Sum of all side

So,

==> 50 = (x+4) + (x) + (2x-6)

==> 4x - 2 = 50

==> 4x = 50 + 2

==> 4x = 52

==> x = 52/4

==> x = 13

Hence

• First Side be (x+4) = 13+4 = 17 m
• Second Side be (x) = 13 m
• Third Side be (2x-6) = 2*13 - 6 = 26-6 = 20m

___________________

Now, Calculate Area of triangle by, Heron's Formula

Let, Here Side of ∆ ABC be

• AB = 17m
• BC = 13m
• CA = 20m

★ Semi perimeter(S) = (AB + BC + CA)/2

So,

==> S = (17+13+20)/2

==>S = 50 / 2

==> S = 25.

Heron's Formula

★ Area of triangle = √[S(S-AB)(S-BC)(S-CA)]

==> Area of triangle = √[25*(25-17)(25-13)(25-20)]

==> Area of triangle = √[25*8*12*5]

==> Area of triangle = √[5*5*4*2*4*3*5]

==> Area of triangle = (5*4).√[2*3*5]

==> Area of triangle = 20 √30 m²

Since

• Area of Triangle be = 20.√30 m²

___________________

Answer 2

Step-by-step explanation:

Let the Length of the smaller side = x cm

Length of the second side = (x + 4) cm

Length of the third side = (2x - 6) cm

Perimeter of a Triangle = 50 cm

According to Question now,

➳ Perimeter of triangle = Sum of all sides

➳ 50 = x + (x + 4) + (2x - 6)

➳ 50 = x + x + 4 + 2x - 6

➳ 50 = 4x - 2

➳ 4x = 50 + 2

➳ x = 52/4

➳ x = 13

Therefore,

Length of the smaller side = x = 13 cm

Length of the second side = (x + 4) = 13 + 4 = 17 cm

Length of the third side = (2x - 6) = 2(13) - 6 = 26 - 6 = 20 cm

_____________________

Now, we will find the Semi Perimeter of triangle :

Let a = 13 cm, b = 17 cm = c = 20 cm

➳ S = a + b + c/2

➳ S = 13 + 17 + 20/2

➳ S = 50/2

➳ Semi Perimeter = 25 cm

Now, we will find the area of triangle by using herons Formula :

$: \implies \sf \: Area = \sqrt{s(s - a) \: (s - b) \: (s - c) }$

$: \implies \sf \: Area = \sqrt{25(25 - 13) \: (25 - 17) \: (25- 20) }$

$: \implies \sf \: Area = \sqrt{25 \times 12 \times 8 \times 5 }$

$: \implies \sf \: Area = \sqrt{12000 }$

$: \implies \underline{ \boxed{ \sf \: Area = 20\sqrt{30 } \: cm^2}}$