Atoy manufacturer needs a piece of plastic in the shape of a right triangle with the longer leg 6cm more than the shorter leg and the hypotenuse 6cm more than the longer leg. how long should the sides of the triangle be?

X= shorter leg
x+6= longer leg
(x+6)+6= hypotenuse

Use the Pythagorean theorem to find the length of each side.

a= one leg
b= other leg
c= hypotenuse.

SHORTER LEG:
a^2 + b^2= c^2
substitute each length above

x^2 + (x+6)^2= [(x+6)+6]^2
square each set of parentheses; add inside right side parentheses before squaring

x^2 + (x+6)(x+6)= (x+12)(x+12)
squaring is multiplying each parentheses by itself one time; use the FOIL method: multiply first, multiply outside, multiply inside, multiply last

x^2 + x^2 + 6x + 6x + 36= x^2 + 12x + 12x + 144
combine like terms

2x^2 + 12x + 36= x^2 + 24x + 144
subtract x^2 from both sides

x^2 + 12x + 36= 24x + 144
subtract 24x from both sides

x^2 - 12x + 36= 144
subtract 144 from both sides

x^2 - 12x - 108= 0
factor

(x + 6)(x - 18)= 0
set each parentheses equal to 0

x + 6= 0
x= -6

x - 18= 0
x= 18 cm shorter leg

***Since a side cannot be a negative number, x= 18. Substitute x=18 to find the other sides.

LONGER LEG:
= x+6
= 18 + 6
= 24 cm longer leg

HYPOTENUSE:
= (x+6)+6
= 18 + 6 + 6
= 30 cm hypotenuse

CHECK:
18^2 + 24^2= 30^2
324 + 576= 900
900= 900

ANSWER:
x= 18 cm shorter leg
x+6= 24 cm longer leg
(x+6)+6= 30 cm hypotenuse

Hope this helps! :)