Question

The hypotenuse of an isosceles right triangle is 9 feet longer than either of its legs. find the exact length of each side

Answer 1

Use the Pythagorean theorem:
a - a legc - a hypotenuse
$c^2=a^2+a^2\\\\c^2=2a^2$
We have:
$c=a+9$
Substitute:
$(a+9)^2=2a^2$
Use (a + b)² = a² + 2ab + b²
$a^2+2\cdot a\cdot 9+9^2=2a^2\\\\a^2+18a+81=2a^2\ \ \ |-2a^2\\\\-a^2+18a+81=0\ \ \ |\cdot(-1)\\\\a^2-18a-81=0\ \ \ \ |+81\\\\a^2-18a=81\\\\a^2-2\cdot a\cdot9=81\ \ \ \ |+9^2\\\\a^2-2\cdot a\cdot9+9^2=81+9^2\\\\(a-9)^2=162\to a-9=\sqrt{162}\ \ \ |+9\\\\a=9+\sqrt{81\cdot2}\\\\a=9+9\sqrt2$
Answer:$9+9\sqrt2;\ 9+9\sqrt2;\ 18+9\sqrt2$