Question

the perimeter of a triangle is 50 cm. one side of the triangle is 4 cm longer than the smallest side and the third side is 6 cm less than twice the smallest side. find the area of the triangle.​

Answer 1

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 :

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

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

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

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

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

Answer 2

Let the triangle Be ABC.

Let smallest side AB = x

one side of the triangle BC= x+4

Third side of the triangle AC= 2x-6

Perimeter of the triangle = AB+BC+AC

But According to the question

Perimeter=50CM

Therefore,

AB+BC+AC=50

x+x+4+2x-6 = 50

4x-2 = 50

4x = 50+2

4x = 52

x = 52/4

x = 13

Now:

AB = 13

BC = 13+4 = 17

AC = 2×13-6 = 20

By heron's formula we get Area

= 20root30CM2