Question

$\overline{hi} \parallel \overline{gj}$. find $fi$.

Explanation:

Step1: Identify similar triangles

Since $\overline{HI} \parallel \overline{GJ}$, $\triangle FHI \sim \triangle FGJ$ by AA similarity.

Step2: Set up proportion of sides

Corresponding sides of similar triangles are proportional. Let $FI = x$, so $JI = x - 17$. The ratio of sides:
$$\frac{FG}{FH} = \frac{FJ}{FI}$$
Substitute values: $FG=16$, $FH=24$, $FJ=x-17$, $FI=x$
$$\frac{16}{24} = \frac{x - 17}{x}$$

Step3: Simplify and solve for x

Simplify $\frac{16}{24}$ to $\frac{2}{3}$:
$$\frac{2}{3} = \frac{x - 17}{x}$$
Cross-multiply:
$$2x = 3(x - 17)$$
Expand right side:
$$2x = 3x - 51$$
Rearrange to solve for $x$:
$$3x - 2x = 51$$
$$x = 51$$

Answer:

51