Atoy manufacturer needs a piece of plastic in the shape of a right triangle with the longer leg 2 cm more than the shorter leg and the hypotenuse 4 cm more than the shorter leg. how long should the sides of the triangle be

Let x be the length of shortest leg of right triangle (in cm)

(x + 2) = length of longer leg

(x + 4) = length of hypotenuse Use c2 = a2 + b2 formula that links the 3 sides of the triangle.

Then:

(x + 4)2 = x^2 + (x + 2)2

(x^2 + 8x + 16)  =  x^2  +  (x^2 + 4x + 4) - then combine terms on the right side

0  =  x2 - 4x - 12 - factor it.

0  =  (x - 6) ( x + 2)

 Try (x + 2) = 0  which implies x = -2 cm, but you know the triangle side can't be negative, so remove this solution. Try this (x - 6)  = 0  which implies x = 6 cm. So your right triangle has sides : 6 cm , 8 cm and 10 cm for the hypotenuse.

Let the length of the shorter leg of the triangle be x cm So, the length of the longer leg is x+ 2 cm the length of the hypotenuse = x+4 cm Pythagorean theorem states that "The square of the hypotenuse is equal to the sum of squares of the other two sides." Therefore, (x+4)^2 = x^2 + (x+2)^2 x^2 + 8x + 16 = x^2 + x^2 +4x + 4 x^2 - 4x - 12=0 Let's factorize using splitting the middle term method, x^2 - 6x + 2x - 12 =0 x(x-6) +2(x-6)=0 (x+2)(x-6)=0 x=-2 or x=6 As we have taken x as the length, it cannot be negative. Therefore x= 6cm Longer side = x+2 =6+2 = 8 cm Hypotenuse = x + 4 = 6+4=10 cm The sides are, 6cm, 8cm and 10 cm.