Question

a toy 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?

Answer 1

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! :)