The Solution.
Representing the given information in a diagram.
Applying Pythagorean Theorem on the right-angled triangle above, we get
[tex]\begin{gathered} (4x+17)^2=x^2+(4x+16)^2 \\ C\text{learing the brackets, we get} \\ 4x(4x+17)+17(4x+17)=x^2+4x(4x+16)+16(4x+16) \end{gathered}[/tex][tex]16x^2+68x+68x+289=x^2+16x^2+64x+64x+256[/tex]Collecting the like terms, we get
[tex]16x^2-16x^2-x^2+136x-128x+289-256=0[/tex][tex]\begin{gathered} -x^2+8x+33=0 \\ \text{Multiplying through by -1, we get} \\ x^2-8x-33=0 \end{gathered}[/tex]Solving quadratically by factorization method, we get
[tex]\begin{gathered} x^2+3x-11x-33=0 \\ (\text{Having substituted +3x and -11x for -8x)} \end{gathered}[/tex][tex]\begin{gathered} x(x+3)-11(x+3)=0 \\ (x+3)(x-11)=0 \end{gathered}[/tex][tex]\begin{gathered} x+3=0\text{ or x-11=0} \\ x=-3\text{ or x = 11} \end{gathered}[/tex]Since the length of a side of a triangle cannot be negative, we discard x = -3.
So, the correct length for the shortest side of the triangle is 11cm
The longer side = 4x+16 = 4(11)+16 = 44+16 = 60cm
The hypotenuse side = 4x+17 = 4(11) +17 = 44+17 =61cm.
Therefore, the correct answer is
11cm, 60cm, and 61cm
