Similar Triangles and Rectangle EFGD: Calculate Perimeter Using 3, 6, and 8 Units

To solve for the perimeter of rectangle $EFGD$, we follow these steps:

Step 1: Use the similarity of the triangles $\triangle BCD \sim \triangle FED$. This implies the sides are proportional:

$\frac{CD}{ED} = \frac{BC}{EF}$

Substitute the known lengths $CD = 8$ and $ED = 3$:

$\frac{8}{3} = \frac{6}{EF}$

Solving for $EF$, cross-multiply:

$8 \times EF = 3 \times 6$ $EF = \frac{18}{8} = 2.25$

Step 2: The length and width of rectangle $EFGD$ are $2.25$ (as calculated from $EF$) and $3$ (equal to $ED$).

Step 3: Calculate the perimeter $P$ of $EFGD$:

$P = 2 \times (EF + ED) = 2 \times (2.25 + 3) = 2 \times 5.25$ $P = 10.5$

The perimeter of rectangle $EFGD$ is $10.5$.