Question

એક ત્રિકોણની પરિમિતિ 32 સેમી અને બાજુ 8 સેમી અને 11 સેમી હોય તો તેનું ક્ષેત્રફળ શોધો ​

Answer 1

Given:

એક ત્રિકોણની પરિમિતિ 32 સેમી અને બાજુ 8 સેમી અને 11 સેમી હોય તો તેનું ક્ષેત્રફળ શોધો ​

Find the area if the perimeter of a triangle is 32 cm and the sides are 8 cm and 11 cm?

To find:

The area

Solution:

Let "a", "b" and "c" represents the sides of the triangle.

Finding the third side of the triangle:

We know,

Perimeter of a triangle = sum of all 3 sides

∴ a + b + c = Perimeter

on substituting a = 8 cm, b = 11 cm and peirmeter = 32 cm

⇒ 8 + 11 + c = 32

⇒ 19 + c = 32

⇒ c = 13 cm

Finding the Semiperimeter of the triangle:

Semi-Perimeter, $S = \frac{Perimeter}{2} = \frac{a+b+c}{2} = \frac{8 + 11+ 13}{2} = \bold{16\:cm}$

Finding the Area of the triangle:

Heron's formula: $\boxed{\bold{Area = \sqrt{S(S-a)(S-b)(S-c)} }}$

Therefore,

The area of the triangle is,

= $\sqrt{16 \times (16-8)\times (16-11)\times (16-13)} }}$

= $\sqrt{16 \times 8\times 5 \times 3} }}$

= $\sqrt{2\times 8 \times 8\times 5 \times 3} }}$

= $8\sqrt{2\times 5 \times 3} }}$

= $\bold{8\sqrt{30} \:cm^2}$

Thus, the area of the triangle is → 8√30 cm².

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