Question

perimeter of right triangle is 12 cm its hypotenuse is 5cm . find length of other two sides.

Answer 1

14 and 143 according to pytha goras theorem

Answer 2

One way is two directly think of a right triangle with hypotenuse 5, which in this case is 3,4,5 triangle.
However, when the length of the hypotenuse is greater, this might not work. So , here is the universal method!!
Let the length of the other two legs be x and y.
Now given that
$x + y + 5 = 12 \\ x + y = 7$
And by Pythagoras Theorem,
${x}^{2} + {y}^{2} = {5}^{2} \\ {x}^{2} + {y}^{2} = 25$
Now consider this,
${(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy \\ {x}^{2} + {y}^{2} = {(x + y)}^{2} - 2xy$
Substitute this to get
${(x + y)}^{2} - 2xy = 25 \\ {7}^{2} - 2xy = 25 \\ 49 - 25 = 2xy \\ 24 = 2xy \\ xy = 12$
Now think of numbers that on adding give 7 and on multiplying give 12.
Clearly such numbers are 3 and 4 and so you have your answer.
Or, once again,
$x + y = 7 \\ y = 7 - x$
Substitute this,
$x(7 - x) = 12 \\ 7x - {x}^{2} = 12 \\ {x - 7x }^{2} + 12 = 0$

Solving this, u get
$(x - 3)(x - 4) = 0 \\ x = 3 \\ x = 4$
Substitute any of these into the xy equation and you have your answer.