Question

In a right angle triangle Perimeter is 60 cm and hypotenus 25 cm find the area of triangle

Answer 1

Given, length of the hypotenuse = 25 cm

perimeter of the right angled triangle = 60 cm

           

⇒ Perimeter of the right angled triangle = length of hypotenuse + length of 1st side + length of 2nd side

⇒ 60 cm = 25 cm + length of 1st side + length of 2nd side

⇒ 60 - 25 cm = length of 1st side + length of 2nd side

⇒ 35 cm =  length of 1st side + length of 2nd side

Let length of 1st side = x

⇒ 35 cm = x + length of 2nd side

⇒ 35 cm - x = length of 2nd side

Now, length of 1st side = x cm

         length of 2nd side = 35 cm - x

As the given triangle is a right angled triangle, we can apply "Pythagoras Theorem".

 Pythagoras Theorem : -

( 1st side )^2 + ( 2nd side )^2 =  hypotenuse^2

∴ ( x )^2 + ( 35 - x )^2 =  ( 25 )^2

⇒ x^2 + ( 35 )^2 + x^2 - 2( 35 * x ) = 625

⇒ x^2 + x^2 + 1225 - 70 x = 625

⇒ 2x^2 - 70x + 1225 - 625 = 0

⇒ 2x^2 - 70x + 600 = 0  

⇒ 2( x^2 - 35 x + 300 ) = 0

⇒ x^2 - 35x + 300 = 0

⇒ x^2 - ( 15 + 20 )x + 300 = 0

⇒ x^2 - 15x - 20x + 300 = 0

⇒ x( x -15 ) - 20( x - 15 ) = 0

⇒ ( x -15 )( x -20 ) = 0

         By Zero Product Rule

x =  15 or  x = 20

∴ As the sum of 15 and 20 is 35, sides of the triangle are 15 cm and 20 cm.

We know, Area of right angled triangle = 1 / 2 x product of its sides(  not hypotenuse )

∴ Area of the triangle = 1 / 2 x 15 x 20 cm^2

⇒ Area of the triangle = 15 x 10 cm^2

⇒ Area of the triangle = 150 cm^2

Therefore, area of the right angled triangle is 150 cm^2

$\:$

Answer 2

LET THE SIDES OF THE TRIANGLE BE a,b,c AND c BE THE HYPOTENUSE

THEN PERIMETER OF TRIANGLE=a+b+c=60

=a+b+25=60

=a+b=35 (EQ-1)

BY PYTHAGORAS THEOREM

a^2+b^2=c^2

(a+b)^2-2(ab)=25^2

35^2-2ab=25^2 (PUTING VALUE OF a+b from eq -1)

2ab= 1225-625

ab=6002

ab=300 cm

NOW AREA OF TRIANGLE =ab2

=150 cm^2