Question

1. The perimeter of a triangle is 50 cm. One side of a triangle is 4 cm longer than the smaller side and the third side is 6 cm less than twice the smaller side. Find the area of the triangle.

Answer 1

Let the smaller side be x cm.
Third side = 6 less than twice smaller number = (2x - 6) cm
Other side = 4 cm longer than smaller side = (x + 4) cm

Perimeter = 50 cm
x + (2x - 6) + (x + 4) = 50
4x - 2 = 50
4x = 52
x = 13 cm

So the 3 sides of the triangle will be as follows:
x = 13 cm
2x - 6 = 2*13 - 6 = 20 cm
x + 4 = 13 + 4 = 17 cm

If we know the 3 sides of a triangle, we can find its area by Heron's formula which is given by:

A = √(s(s-a)(s-b)(s-c)) where a,b,c are sides of triangle and s is the semi-perimeter of the triangle given by s = (a+b+c)/2

I guess, from here it is easy to find the value of S and substitute in the Heron's formula and find the area A.

Let me know if you need any help.

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 :

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

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

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

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

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