Question

the length of the longer leg of a right triangle is 3 ft more than three times the length of the shorter leg. the length of the hypotenuse is 4 ft more than three times the length of the shorter leg. find the side lengths of the triangle.

Answer 1

with the pythagorean theorem

`\begin{gathered} (4+3x)^2=x^2+(3+3x)^2 \\ 16+24x+9x^2=x^2+9+18x+9x^2 \\ 16+24x+9x^2=10x^2+18x+9 \\ 16+24x+9x^2-9=10x^2+18x+9-9 \\ 9x^2+24x+7=10x^2+18x \\ 9x^2+24x+7-18x=10x^2+18x-18x \\ 9x^2+6x+7=10x^2 \\ 9x^2+6x+7-10x^2=10x^2-10x^2 \\ -x^2+6x+7=0 \end{gathered}`

using the formula of the quadratic equation

`\begin{gathered} x_(1,\: 2)=(-b\pm√(b^2-4ac))/(2a) \\ x1=(-6+√(6^2-4\left(-1\right)\cdot\:7))/(2\left(-1\right))=-1 \\ x2=(-6-√(6^2-4\left(-1\right)\cdot\:7))/(2\left(-1\right))=7 \end{gathered}`

the length cannot be negative, therefore x=7

length of the shorter leg is: 7ft

length of the longer leg is: 3+3(7)= 24ft

length of the hypotenuse is: 4+3(7)= 25ft