Question

what is the area? write your answer as a fraction or as a whole or mixed number. \\(\frac{2}{3}\\) in \\(\frac{2}{3}\\) in 1 in square inches submit

Explanation:

Step1: Identify the shape and formula

The shape is a trapezoid. The formula for the area of a trapezoid is $A = \frac{1}{2}(b_1 + b_2)h$, where $b_1$ and $b_2$ are the lengths of the two parallel sides (bases), and $h$ is the height.
Here, $b_1 = \frac{2}{3}$ in, $b_2 = 1$ in, and $h = \frac{2}{3}$ in.

Step2: Substitute values into the formula

First, find the sum of the bases: $b_1 + b_2 = \frac{2}{3} + 1 = \frac{2}{3} + \frac{3}{3} = \frac{5}{3}$ in.
Then, multiply by the height and $\frac{1}{2}$: $A = \frac{1}{2} \times \frac{5}{3} \times \frac{2}{3}$.

Step3: Simplify the expression

The 2 in the numerator and the 2 in the denominator (from $\frac{1}{2}$ and $\frac{2}{3}$) cancel out: $A = \frac{1}{1} \times \frac{5}{3} \times \frac{1}{3} = \frac{5}{9}$? Wait, no, wait. Wait, let's recalculate. Wait, $\frac{1}{2} \times \frac{5}{3} \times \frac{2}{3}$: the 2 in $\frac{1}{2}$ and the 2 in $\frac{2}{3}$: $\frac{1}{2} \times \frac{2}{3} = \frac{1}{3}$, then $\frac{1}{3} \times \frac{5}{3} = \frac{5}{9}$? Wait, no, that's wrong. Wait, no, the height is $\frac{2}{3}$, and the bases are $\frac{2}{3}$ and 1. Wait, maybe I mixed up the bases. Wait, the top base is $\frac{2}{3}$, the bottom base is 1, and the height is $\frac{2}{3}$. So $A = \frac{1}{2}(\frac{2}{3} + 1)\times\frac{2}{3}$. Let's compute $\frac{2}{3} + 1 = \frac{5}{3}$. Then $\frac{1}{2} \times \frac{5}{3} \times \frac{2}{3}$. The $\frac{1}{2}$ and $\frac{2}{3}$: multiply numerators: 1×2=2, denominators: 2×3=6, so $\frac{2}{6} = \frac{1}{3}$. Then $\frac{1}{3} \times \frac{5}{3} = \frac{5}{9}$? Wait, no, that can't be. Wait, maybe the shape is a parallelogram? Wait, no, the dashed line is the height, and the two bases are parallel. Wait, maybe I misidentified the shape. Wait, if the two parallel sides are both $\frac{2}{3}$ and 1, and the height is $\frac{2}{3}$, but maybe it's a trapezoid. Wait, alternatively, maybe it's a parallelogram? No, the sides are not equal. Wait, no, the formula for trapezoid is correct. Wait, let's do the calculation again. $\frac{1}{2} \times (\frac{2}{3} + 1) \times \frac{2}{3}$. $\frac{2}{3} + 1 = \frac{5}{3}$. Then $\frac{1}{2} \times \frac{5}{3} = \frac{5}{6}$. Then $\frac{5}{6} \times \frac{2}{3} = \frac{10}{18} = \frac{5}{9}$. Wait, that's correct? Wait, no, wait, $\frac{5}{6} \times \frac{2}{3}$: 5×2=10, 6×3=18, simplify to $\frac{5}{9}$. Wait, but let's check again. Wait, maybe the shape is a trapezoid with bases $\frac{2}{3}$ and 1, height $\frac{2}{3}$. So area is average of the bases times height: $(\frac{\frac{2}{3} + 1}{2}) \times \frac{2}{3} = (\frac{5}{3} \div 2) \times \frac{2}{3} = (\frac{5}{6}) \times \frac{2}{3} = \frac{10}{18} = \frac{5}{9}$. Yes, that's correct.

Answer:

$\frac{5}{9}$