The problem is about finding the leg lengths of an isosceles right triangle that has an area of 64 square feet. Let's analyze the equation and solve it step by step.

1. The area of a triangle is given by: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ For an isosceles right triangle, the base and height are equal, so if we let $x$ represent the length of each leg, the area becomes: $\frac{1}{2} x^2 = 64$

2. Solving for $x$: $x^2 = 64 \times 2 = 128$ $x = \sqrt{128} = \pm \sqrt{64 \times 2} = \pm 8\sqrt{2}$

Since the leg length must be positive, we use $x = 8\sqrt{2}$.

Drop-down Selections

1. Solutions to the system: $x = 8\sqrt{2}$ and $x = -8\sqrt{2}$
2. Length of one leg of the sun shade: $8\sqrt{2}$ feet

Would you like further details on the calculation, or do you have additional questions?

Here are some related questions for deeper understanding:

1. How do you solve for the area of other types of triangles, such as equilateral or scalene?
2. What is the significance of restricting solutions to positive values in geometry?
3. How can the Pythagorean Theorem be used to verify the lengths of the legs in this isosceles triangle?
4. What other methods exist to solve quadratic equations beyond taking square roots?
5. How would the solution change if the triangle were not isosceles?

Tip: Always double-check the units in geometry problems to ensure consistency throughout the calculations.