Question

a quadrilateral with side lengths \\(\frac{2}{3}\\) in., \\(\frac{5}{6}\\) in., \\(\frac{2}{3}\\) in., and \\(\frac{1}{2}\\) in. (the quadrilateral is shaded).

Explanation:

Step1: Identify the shape

The figure is a quadrilateral with two pairs of equal sides ($\frac{2}{3}$ in, $\frac{2}{3}$ in and $\frac{1}{2}$ in, $\frac{5}{6}$ in? Wait, no, actually, looking at the sides: two sides are $\frac{2}{3}$ in, one is $\frac{1}{2}$ in, one is $\frac{5}{6}$ in. Wait, maybe it's a trapezoid? Wait, no, maybe it's a quadrilateral where we need to find the perimeter. Perimeter is the sum of all sides.

Step2: List the side lengths

The side lengths are $\frac{2}{3}$ in, $\frac{5}{6}$ in, $\frac{2}{3}$ in, and $\frac{1}{2}$ in.

Step3: Convert to common denominator

To add fractions, find a common denominator. The denominators are 3, 6, 3, 2. The least common denominator is 6.

Convert each fraction:

• $\frac{2}{3} = \frac{2\times2}{3\times2} = \frac{4}{6}$
• $\frac{5}{6}$ remains $\frac{5}{6}$
• $\frac{2}{3} = \frac{4}{6}$
• $\frac{1}{2} = \frac{1\times3}{2\times3} = \frac{3}{6}$

Step4: Sum the fractions

Add the numerators: $4 + 5 + 4 + 3 = 16$. The denominator is 6. So $\frac{16}{6}$.

Step5: Simplify the fraction

$\frac{16}{6} = \frac{8}{3} = 2\frac{2}{3}$ inches. Wait, but maybe the shape is a parallelogram? Wait, in a parallelogram, opposite sides are equal. Wait, the sides: $\frac{2}{3}$ and $\frac{2}{3}$ (opposite), and $\frac{1}{2}$ and $\frac{5}{6}$? Wait, no, $\frac{1}{2}$ is $\frac{3}{6}$, $\frac{5}{6}$ is $\frac{5}{6}$. So maybe it's not a parallelogram. But regardless, perimeter is sum of all sides.

Wait, maybe I misread the sides. Let me check again. The figure: top left is $\frac{2}{3}$ in, top right is $\frac{5}{6}$ in, bottom right is $\frac{2}{3}$ in, bottom left is $\frac{1}{2}$ in. So the four sides are $\frac{2}{3}$, $\frac{5}{6}$, $\frac{2}{3}$, $\frac{1}{2}$.

So sum: $\frac{2}{3} + \frac{5}{6} + \frac{2}{3} + \frac{1}{2}$

Convert to sixths:

$\frac{4}{6} + \frac{5}{6} + \frac{4}{6} + \frac{3}{6} = \frac{4 + 5 + 4 + 3}{6} = \frac{16}{6} = \frac{8}{3} = 2\frac{2}{3}$ inches.

Wait, but maybe the problem is to find the perimeter. Let's confirm.

Answer:

The perimeter of the quadrilateral is $\frac{8}{3}$ inches or $2\frac{2}{3}$ inches.