Question 81476
QUESTION:


In a right triangle, one leg is 7 more than the other. the hypotenuse is 17. find both legs.


ANSWER:



Given ,


Hypotenuse = 17


Assume that the third side is 'x'


Then the second side is = x + 7


By pythagorean theorem,


(hypotenuse)^2 = (thrid  side)^2 + (second side)^2



==> 17^2 = x^2 + (x+7)^2


==> 289 = x^2 + x^2 + 14x + 49



==> 289 = 2x^2 + 14x + 49



Subtract 289 from both sides





Hypotenuse = 17


Assume that the third side is 'x'


Then the second side is = x + 7


By pythagorean theorem,


(hypotenuse)^2 = (thrid  side)^2 + (second side)^2



==> 17^2 = x^2 + (x+7)^2


==> 289 = x^2 + x^2 + 14x + 49



==> 289 - 289 = 2x^2 + 14x + 49 - 289



==> 0 = 2x^2 + 14x- 240



==>  2x^2 + 14x- 240 = 0 




This is a  quadratic equation and can be solved by quadratic formula,

{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}


Comparing with the general quadratic equation, ax^2 + bx + c = 0



we have, a = 2, b = 14 and c = -240




so,

{{{x = (-14 +- sqrt( 14^2-4*2*(-240) ))/(2*2) }}}




{{{x = (-14 +- sqrt( 196 + 1920))/(4) }}}




{{{x = (-14 +- sqrt( 2116))/(4) }}}





{{{x = (-14 +- 46) /(4) }}}




==> x = (-14 + 46) /4 or x = (-14 - 46)/4




==> x = 32/4 or x = -60/4




==> x = 8  or x = -15



Since negative value is not admissible for the side of a triangle,

we can take the thirds side = 8


Then the second side = x + 7 = 8 + 7 = 15



so the sides of the triangle are, 8, 15 and 17.




Hope you found the explanation useful.



Regards.



Praseena.