Quadrilateral BCDE is similar to quadrilateral FGHI. Find the measure of side HI. Figures are not drawn to scale.

Since quadrilateral BCDE is similar to quadrilateral FGHI, the corresponding sides are proportional. Using this information, we can set up the following proportion:

$\frac{BC}{FG}=\frac{DE}{HI}$

We are given that $DE=20$, so we can substitute the value into the equation:

$\frac{BC}{FG}=\frac{20}{HI}$

We are also given that $DC=10$, so we can substitute this value into the equation:

$\frac{10+BC}{FG}=\frac{20}{HI}$

We are given that $HG=4.3$, so we can substitute this value into the equation:

$\frac{10+BC}{4.3}=\frac{20}{HI}$

To solve for HI, we can cross-multiply and solve for HI:

$HI(10+BC)=20(4.3)$

Divide both sides by $10+BC$:

$HI=\frac{20(4.3)}{10+BC}$

We cannot solve for the measure of side HI without knowing the measure of side BC.

The measurement to BC is blank

Apologies for the confusion. Without knowing the measurement of side BC, we cannot determine the exact measure of side HI. The given information and the given proportions are not sufficient to find the measure of HI.