Right Triangle Area

The sum of the lengths of the three sides of a right triangle is 18. The sum of the squares of the lengths of the three sides is 128. What is the area of this right triangle?

Source: mathcontest.olemiss.edu 3/5/2012

SOLUTION
$a+b+c=18\qquad\qquad\left (1\right )$
$a^2+b^2+c^2=128\qquad\left (2\right )$

By the Pythagorean theorem
$a^2+b^2=c^2$
Substitute the value of $a^2+b^2$ into Eq. (2)
$c^2+c^2=128$
$2c^2=128$
$c^2=64$
$c=\pm\sqrt{64}$
$c=8$
Substitute the value of $c$ into Eq. (1)
$a+b+8=18$
$a+b=10$
$\left (a+b\right )^2=100$
$a^2+2ab+b^2=100$
Substitute the value of $a^2+b^2$
$2ab+a^2+b^2=100$
$2ab+64=100$
$2ab=36$
Divide both sides by 4
$\frac{ab}{2}=9$
Area of right triangle = $9$

Answer: $9$.