Answer: The side lengths of the triangle are: 13 (short leg), 85 (hypotenuse), 84 (long leg)
Step-by-step explanation:
We have a right triangle with the following given lengths:
Shorter leg: [tex]x[/tex]
Longer leg: [tex]19 ft+5x[/tex]
Hypotenuse: [tex]20 ft+5x[/tex]
Since this is a right triangle, we can use the Pithagorean theorem:
[tex](Hypotenuse)^{2}=(Shorter-leg)^{2}+(Longer-leg)^{2}[/tex]
[tex](20 ft+5x)^{2}=x^{2}+(19 ft+5x)^{2}[/tex]
Solving the parenthesis:
[tex](20 ft)^{2}+(2)(20 ft)(5x)+(5x)^{2}=x^{2}+(19 ft)^{2}+(2)(19 ft)(5x)+(5x)^{2}[/tex]
[tex]39 ft^{2}+10 ft x-x^{2}=0[/tex]
Multiplying the equation by (-1):
[tex]x^{2}-10 ft x-39ft^{2}=0[/tex]
Now we have this quadratic equation, which can be solved with the quadratic formula [tex]x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}[/tex]
Where [tex]a=1[/tex], [tex]b=-10[/tex], [tex]c=-39[/tex]
Substituting the known values and choosing the positive result of the equation:
[tex]x=\frac{-(-10)\pm\sqrt{(-10)^{2}-4(1)(-39)}}{2(1)}[/tex]
[tex]x=13 ft[/tex]
Now we can fin the measure of each leg:
Shorter leg: [tex]x=13 ft[/tex]
Longer leg: [tex]19 ft+5(13 ft)=84 ft[/tex]
Hypotenuse: [tex]20 ft+5(13 ft)=85 ft[/tex]
