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.

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About mvtrinh

Retired high school math teacher.
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