Answer
$50$ yards, $120$ yards and $130$ yards.
Work Step by Step
The lengths of the legs are $2x+20$ and $x$.
The length of the hypotenuse is $2x+30$.
Apply the Pythagorean Theorem.
$leg_1^2+leg_2^2=hypotenuse^2$
$\Rightarrow (2x+20)^2+x^2=(2x+30)^2$
Square $(2x+20)$ and $(2x+30)$.
$\Rightarrow 4x^2+80x+400+x^2=4x^2+120x+900$
Subtract $4x^2+120x+900$ from both sides.
$\Rightarrow 4x^2+80x+400+x^2-4x^2-120x-900=4x^2+120x+900-4x^2-120x-900$
Simplify.
$\Rightarrow x^2-40x-500=0$
Rewrite the middle term $-40x$ as $-50x+10x$.
$\Rightarrow x^2-50x+10x-500=0$
Group terms.
$\Rightarrow (x^2-50x)+(10x-500)=0$
Factor each group.
$\Rightarrow x(x-50)+10(x-50)=0$
Factor out $(x-50)$.
$\Rightarrow (x-50)(x+10)=0$
Set each factor equal to zero.
$x-50=0$ or $x+10=0$
Isolate $x$.
$x=50$ or $x=-10$
Take the positive value because $x$ represents a dimension and it has to be positive.
$x=50$ yards
Length of the other sides are.
$\Rightarrow 2x+20 = 2(50)+20=120$ yards.
$\Rightarrow 2x+30 = 2(50)+30=130$ yards.
