Question

the sides of triangle are in A.P. the area and the perimeter of a triangle are numerically equal. find its peeimeter . any one give answer in first 5 minutes i will mark the that answer as brainliest . please dont give aase vase answer​

Answer 1

⊕ANSWER⊕

Since there are three sides in a ∆, so let the terms of A.P be

a - d, a, a + d

Here, first term = smallest side = a - d

common difference = d

So the perimeter = sum of sides

= a - d + a + a + d

= 3a

Area for right angled triangle = 1/2 × base × height.

Since, a + d would be the greatest, so it would be hypotenuse. Hence, other two sides (a -d and a) would be base and height.

So area = 1/2 × (a - d) × a

= (a - d)(a)/2

Given that perimeter = area

Perimeter = 3a

and area = (a - d)(a)/2

=> $3a =\frac{(a-d)(a)}{2} \\a-d = 3a * \frac{2}{a} \\a-d = 6$

Now, a + d would be equal to $\sqrt{a^{2} - (a-d)^{2} }$

since its a right angled triangle and a + d is hypotenuse.

=> $(a+d)^{2} = a^{2} + (a-d)^{2} \\(a+d)^{2} = a^{2} + 6^{2} \\(a-d)^{2} + 4ad = a^{2} + 36\\4ad = a^{2} \\4d = a$

[ Since,

(a + d)² = (a - d)² + 4ad

and, (a - d) = 6

=> (a - d)² = 36 ]

So we have

a - d = 6

and 4d = a

So put the value of a

=> 4d - d = 6

=> 3d = 6

=> d = 6/3

=> d = 2

So a - d = 6

=> a - 2 = 6

=> a = 6 + 2

=> a = 8

So perimeter = 3a = 3(8) = 24

Answer :- 24