Question

Find the area of a triangle having perimeter 32cm. One side of its side is equal to 11cm and difference of the other two is 5cm.

Answer 1

Given :-

perimeter of the triangle = 32cm

» semi-perimeter = 32/2 = 16cm

one of the side of the triangle = 11cm

ATQ, the difference between other two sides of the triangle is 5cm.

let one of them be x.

therefore the other one = x - 5

perimeter of the triangle = sum of all sides

➡ 11 + x + (x - 5) = 32cm

➡ 11 + x + x - 5 = 32cm

➡ 6 + 2x = 32cm

➡ 2x = 26

➡ x = 26/2 = 13cm

the sides are :-

• x = 13cm

• x - 5 = 13 - 5 = 8cm

now, we've to find the area of the triangle. since all sides are given but height isn't given.

so we'll use heron's formula which is √s(s - a)(s - b)(s - c)

where s is the semi-perimeter of the triangle and a,b and c are the sides of the triangle respectively.

therefore it's area = √[16(16 - 11)(16 - 13) (16 - 8)]

= √(16 × 5 × 3 × 8)

= √(2 × 2 × 2 × 2 × 5 × 3 × 2 × 2 × 2)

= √(2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 5)

= 2 × 2 × 2√(2 × 3 × 5)

= 8√30cm²

hence, it's area is 20√6cm²

Answer 2

Answer :-

Area of triangle = 8√30

--------------------

Given :-

• Perimeter = 32cm
• one side = 11cm
• Difference of other 2 sides - 5cm

To Find :-

• Area of triangle

Solution :-

Given that difference of two sides = 5

Let one side be = x

So, x - y = 5

.·. 3rd side = y = x - 5

-----------------

Perimeter = 32cm

11 + x + x - 5 = 32

11 + 2x - 5 = 32

2x  = 32 - 11 + 5

2x = 26

x = 26/2

x = 13

--------------

Finding the sides of triangle :-

One side = 11cm

Second side = x = 13cm

Third side = x - 5 = 8cm

------------

Finding the area using Heron's formula :-

Heron's formula ---

$\sqrt{s(s-a)(s-b)(s-c)}$

Here,

S = Perimeter/2

• S = 32/2 = 16
• a = 11
• b = 13
• c = 8

Substituting in formula :-

→ $\sqrt{16(16-11)(16-13)(16-8)}$

→ $\sqrt{16(5)(3)(8)}$

→ $\sqrt{2*2*2*2*5*3*2*2*2}$

→ 2 x 2 x 2 √5 x 3 x 2

→ 8√30