Answer
See graph and explanations.
Work Step by Step
Step 1. Draw a diagram as shown, the area of the triangle enclosed by the line segment is given by $A=\frac{1}{2}ab$
Step 2. Given the length of the line segment as $20$, we have $a^2+b^2=20^2$
Step 3. Take the derivative of the area with respect to $a$ to get $\frac{dA}{da}=\frac{1}{2}b+\frac{1}{2}a\frac{db}{da}$
Step 4. Take the derivative of the equation in step 2 with respect to $a$ to get $2a+2b\frac{db}{da}=0$, which gives $\frac{db}{da}=-\frac{a}{b}$
Step 5. The result from step 3 becomes $\frac{dA}{da}=\frac{1}{2}b-\frac{1}{2}a\frac{a}{b}=\frac{b^2-a^2}{2b}$
Step 6. The area will reach an extreme when $\frac{dA}{da}=0$, which gives $a=b$
Step 7. At the end points, when $a\to0, A\to0$ and $a\to20,b\to0,A\to0$, we know the extreme is a maximum. Thus the area reaches a maximum when $a=b$.
