The sum of the length of the three sides of a right triangle is 26. The sum of the squares of the lengths of the three sides is 288. Determine the area of the triangle.
+2666 Let \(a\) and \(b\) be the legs of the triangle, and let \(c\) be the hypotenuse. From the Pythagorean Theorem, along with the given information, we have the following system of equations:
\(a^2 + b^2 = c^2\) (1)
\(a+b+c=26\) (2)
\(a^2+b^2+c^2=288\) (3)
Subsituting (1) in (3) gives us: \(2c^2 =288\). Solving this equations shows that \(c = 12 \)
Now, we have the system:
\(a^2 + b^2 = 144\) (1)
\(a+b = 14\) (2)
Recall the identity: \(a^2 + b^2 = (a+b)^2-2ab\)
This means we have the equation: \((a+b)^2-2ab=144\)
Now subsituting in (2) gives us: \(14^2 - 2ab = 144\)
Can you solve for the area from here?
Hint: the area, because it is a right triangle, is \({ab \over 2}\)
