Question

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

Answer 1

Heya !!!

∆ABC is a triangle.

Let me smaller side be X cm.

One side = (X +4) cm.

And,

Third side =( 2X -6). cm.

Perimeter of triangle = 50 cm

X + X +4 + 2X -6 = 50 cm

4X - 2 = 50

4X = 50+2

4X = 52

X = 52/4

X = 13 cm

And,

Smaller side = X = 13 cm

One side = (X +4) = (13+4) = 17 cm

And,

Third side = (2X -6) cm = ( 2 × 13 -6) = (26-6) = 20 cm

Required sides are 13 , 17 and 20 cm.


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Since,

All sides of a triangle are different it means ∆ABC is an isosceles triangle.

By applying herons formula we can find the area of ∆ABC.

Let AB is a smaller side ,BC is base and AC is longer side.

AB = 13 cm

AC = 20 cm

And,

BC = 17 cm

Semi Perimeter ( S) = 1/2 × ( AB + BC + AC)

=> 1/2 × ( 13 + 17 + 20 )

=> 1/2 × 50

=> 25 cm

Therefore,

Area of ∆ABC = ✓S ( S -A) (S-B) (S-C)

=> ✓25 ( 25 - 13) ( 25 -17 ) ( 25 - 20)

=> ✓25 ( 12) ( 8) ( 5)

=> ✓25 × 12 × 8 × 5

=> ✓5 × 5 × 2 × 2 × 3 × 2 × 2 × 2 × 5

=> 5 × 2 × 2 ✓ 5 × 3 × 2 cm²


=> 20✓30 cm².

Answer 2

Answer:

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

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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 :

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

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

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

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

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