Question

Perimeter of a right angle triangle is 12 cm and its area is 6 CM square. find the side of the triangle

Answer 1

Well, that's a dumb answer.

Since triangle is right-angled, let the 2 legs forming the right angle
be x and y
given : (1/2)xy = 6 ----> xy = 12
let the hypotenuse be z
then z^2 = x^2 + y^2 = x^2 + y^2 + 2xy - 2xy
= (x+y)^2 - 24

we also know that x + y + z = 12
z = 12 - (x+y)
square both sides:
z^2 = 144 - 24(x+y) + x^2 + y^2 + 2xy, but z^2 = x^2 + y^2
x^2+y^2 = 144 - 24(x+y) + x^2+y^2 + 24
24(x+y) = 168
x + y = 7, and xy = 12
just by observation we can see that
x = 3, y = 5 work, as will x = 4, y = 3

So the triangle is 3, 4, 5, the most famous and basic right-angled triangle.
(most mathminded people would have seen that immediately. )

Answer 2

Let the sides of the triangle be x, y, and z.

x + y + z = 12

xy/2 = 6

x + y = 12 - z

xy/2 = 6

(12 - z)z/2 = 6

144 - 12z + z^2/2 = 6

z^2 - 12z + 12 = 0

z = 6 ± √36

z = 6 ± 6

z = 12 or 0

If z = 12, then x + y = 0, which is not possible.

Therefore, z = 0

x + y = 12

xy/2 = 6

xy = 12

x = 12/y

Substitute x = 12/y in x + y = 12

12/y + y = 12

y^2 - 12y + 12 = 0

y = 6 ± √36

y = 6 ± 6

y = 12 or 0

If y = 12, then x = 0, which is not possible.

Therefore, y = 0

x = 12/0

x = undefined

Therefore, the sides of the triangle are 0, 0, and 12 cm.