Question

Here is a Question for The Brainiest Student //___ The sides of a right angled triangle are in A.P. The area and perimeter of the triangles are numerically equal. Find it's perimeter.// No spamming otherwise reported.

Answer 1

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

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


Now, a + d would be equal to
$\sqrt{ {a}^{2} + (a - d) {}^{2} }$
since its a right angled triangle and a + d is hypotenuse.

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

[ 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


Hope you understand

Answer 2

______Heyy Buddy ❤______

____Here's your Answer_______

Let the sides be a-b , a, and a + b with a + b as the Hypotenuse.

Therefore,

PERIMETER of TRIANGLE = (a- b) + a + (a +b)

=> a - b + a + a + b

=> 3a.-------------(1)

Now, Since it is a right angle triangle.

Therefore,

Area of Triangle = 1/2 × Base × Height

=> 1/2 × (a) × (a-b)

=> [ a( a - b)]/2 -----------(2)


A.T.Q.

PERIMETER = AREA

=> 3a = [ a ( a - b)] /2

=> 3a × 2 = a ( a - b)

=> [3a × 2] / a = a- b

=> 6 = a - b. ----------(3)

Now,

By Pythagoras Theorem,

$= > \: {(a + d)}^{2} = {a}^{2} + {(a - b)}^{2}$

Putting eq 3 here, we get...

$= > \: {(a + d)}^{2} = {a}^{2} + {6}^{2}$
Here, ( a + b)^2 = (a-b)^2 + 4ab

=> ( a - b)^2 + 4ab = a^2 + 36

=> putting eq 3.

=> 36 + 4ab = a^2 + 36

=> 4ab = a^2

=> 4ab/a = a

=> a = 4b.

Now,

a - b = 6

=> 4b - b = 6

=> 3b = 6

=> b = 2.

Putting b = 2 in eq 3. we get,

=> 6 = a - b

=> 6 = a - 2

=> a = 8.

Therefore, side of the triangle = 8.

Now,

PERIMETER of TRIANGLE = 3 × sides ( from eq 1)

=> PERIMETER = 3 × 8

=> PERIMETER = 24cm.
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