Answer
$3000$yd
Work Step by Step
We can use the pythagorean theorem, $a^2+b^2=c^2$, to find the lengths of the triangular walkway
$x^2+(x+700)^2=(x+800)^2$
$x^2+x^2+1400x+490000=x^2+1600x+640000$
Combine like terms on the left side in standard quadratic equation form, $ax^2 + bx +c=0$
$x^2-200x-150000=0$
where $a=1$, $b=-200$, and $c=-150000$
now, apply the quadratic formula: $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
$x=\dfrac{(-)(-200)\pm\sqrt{(-200)^2-4(1)(-150000)}}{2(1)}$
$x=\dfrac{200\pm\sqrt{40000+600000}}{2}$
$x=\dfrac{200\pm\sqrt{640000}}{2}$
$x=\dfrac{200\pm\sqrt{640000}}{2}$
$x=\dfrac{200\pm800}{2}$
$x=\dfrac{200+800}{2}$ or $x=\dfrac{200-800}{2}$
$x=\dfrac{1000}{2}$ or $x=\dfrac{-600}{2}$
$x=500$ or $x=-300$
Since the triangle can't have a negative length, $x=500$
The total length of the triangular walkway is
$500$yd$+(500+700)$yd$+(500+800)$yd$=3000$yd
