The area of a triangle is given by the formula:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

We are told that the area of the triangle is $\frac{10}{9}$ square feet. Let's assume the base of the triangle is $b$ feet and the height is $h = \frac{4}{3}$ feet, as given in the problem.

Using the area formula:

$\frac{1}{2} \times b \times \frac{4}{3} = \frac{10}{9}$

Now, solve for $b$:

$b \times \frac{4}{3} = 2 \times \frac{10}{9}$

$b \times \frac{4}{3} = \frac{20}{9}$

Multiply both sides by the reciprocal of $\frac{4}{3}$, which is $\frac{3}{4}$:

$b = \frac{20}{9} \times \frac{3}{4}$

Simplifying:

$b = \frac{60}{36} = \frac{5}{3}$

Thus, the length of the base $b$ is $\frac{5}{3}$ feet.

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Would you like more details on any step of the process or have any questions?

Here are some related questions:

1. How would the solution change if the height were different?
2. What happens to the area of a triangle if both the base and height are doubled?
3. How do you find the area of a triangle when you only know the three sides?
4. How would you express the area of this triangle as a decimal instead of a fraction?
5. How can you verify that the fraction $\frac{5}{3}$ is in its simplest form?

Tip: Always cross-check units (like feet, meters) when solving geometry problems to avoid confusion!